Cellular-resolution optogenetics reveals attenuation-by-suppression in visual cortical neurons

Significance

How neurons transform inputs into outputs is a fundamental building block of brain computation. Here, we measure neurons’ IO functions in the awake and intact brain, where ongoing network activity influences neurons’ responses to input. Using state-of-the-art optogenetic methods to deliver precise inputs to neurons’ cell bodies, or somas, we find a supralinear-to-linear IO function, contrary to previous findings of threshold-linear, strongly saturating, or power law IO functions. This supralinear-to-linear somatic IO function shape allows neurons to decrease responses to, or filter, inputs while they are suppressed below resting firing rates, a computation we term attenuation-by-suppression.

Abstract

The relationship between neurons’ input and spiking output is central to brain computation. Studies in vitro and in anesthetized animals suggest that nonlinearities emerge in cells’ input–output (IO; activation) functions as network activity increases, yet how neurons transform inputs in vivo has been unclear. Here, we characterize cortical principal neurons’ activation functions in awake mice using two-photon optogenetics. We deliver fixed inputs at the soma while neurons’ activity varies with sensory stimuli. We find that responses to fixed optogenetic input are nearly unchanged as neurons are excited, reflecting a linear response regime above neurons’ resting point. In contrast, responses are dramatically attenuated by suppression. This attenuation is a powerful means to filter inputs arriving to suppressed cells, privileging other inputs arriving to excited neurons. These results have two major implications. First, somatic neural activation functions in vivo accord with the activation functions used in recent machine learning systems. Second, neurons’ IO functions can filter sensory inputs—not only do sensory stimuli change neurons’ spiking outputs, but these changes also affect responses to input, attenuating responses to some inputs while leaving others unchanged.

Neurons transform their synaptic inputs into output activity. The way in which neurons, and connected networks of neurons, achieve these transformations is a central and critical component of neural computation. A frequency–current (f-I) curve describes how aggregate synaptic input into a given cell generates spiking. F-I curves (in different contexts called static nonlinearities, neural transfer functions (1–3), or what we will use in this work, input–output or IO functions) thus govern how spikes are produced and passed down to postsynaptic partners. At the network level, the shape of the input–output function affects performance of the overall network. For example, IO function shape affects how well sensory networks can encode features of the world (4). Machine learning (ML) work has confirmed that the shape of IO functions impacts network function, in part by exerting influence on learning speed and accuracy. For example, now-classic work has found that rectified-linear (ReLU) IO functions (or activation functions, the name often used in the ML context) can make learning more efficient and robust than learning with sigmoidal activation functions (5–7).

While work in all these fields often assumes for simplicity that neural IO functions are fixed or static, in real biological networks, IO function shape is critically affected by all the inputs neurons receive—that is, IO function shape is affected by activity in the local recurrent network (8–17). Thus, accurately determining neural IO functions requires measurements in vivo during normal network operation.

Past work has reported various shapes for neural IO functions. In vitro studies, where network activity levels can be low, have often found ReLU (also called threshold-linear) functions (2, 8, 18–21). Studies done in networks with higher levels of network activity than expected in vitro—in anesthetized animals, ex vivo, or in vitro with simulated noise—have found power-law functions, with a supralinear regime throughout the input range (22–27). On the other hand, theoretical studies of integrate-and-fire networks of active neurons have found an analytical description of the IO function in those networks, the Ricciardi input–output function, which has a supralinear regime followed by a linear, then a sublinear regime (28, 29).

Those different descriptions conflict in important ways, and therefore, the IO function shape seen in vivo during awake states of normal operation has been unclear (30–33). This conflict arises in part because electrophysiological studies with control of neural input are difficult to perform in awake animals (34–36). Here, we use two-photon optogenetics to expand measurements of IO functions to hundreds of neurons in awake mice. We deliver the same optogenetic input to cells at different activity levels and thus determine neurons’ input–output responses at different points along their IO function. We deliver input to the soma of cells using a somatically targeted opsin. Thus, here, we study the somatic IO function—how aggregated inputs from the dendrites, or dendritic outputs, affect spike rates.

We use this approach to explore the somatic IO functions of neurons in the visual cortex. We stimulate selected neurons optogenetically, deliver visual stimuli that excite or suppress cells, and measure differences in response to the same optogenetic stimuli. We find that neurons in vivo have an IO function that resembles not a ReLU or a power law function, but the Ricciardi function that is created by integrate-and-fire networks. The measured IO functions, as input varies from low to high, have a supralinearity followed by a linear region, and show some saturation at the upper end of the range of normal input. That upper range is defined by normal network activation levels—neurons’ firing in response to strong, high-contrast visual stimuli. A key observation of this work is that neurons rest near the top of the supralinear region. This means that suppressed neurons produce smaller responses to the same input, suggesting that cortical neurons attenuate responses to input that arrive to suppressed cells. This attenuation-by-suppression may be a computational means by which neurons filter or mask out certain inputs.

Results

Uncovering Input–Output Nonlinearities via Cell-Specific Optogenetic Stimulation.

We use cell-specific two-photon optogenetic stimulation (37–39) to measure cells’ IO functions. By providing the same fixed input at different points along a cell’s IO function, the corresponding evoked changes in output can be used to infer the IO function shape (Fig. 1A). For example, if a cell’s IO function were threshold-linear (Fig. 1 B, Left), the response to a fixed optogenetic input would remain unchanged as the cell’s activity level increases. Similarly, when that cell is suppressed below the threshold, input would elicit no response. Our stimulation bypasses dendritic nonlinearities, by delivering input to the cell at the soma (using a soma-targeted opsin, stChrimsonR), allowing us to measure the transformation of the cell’s aggregate input into output spiking.

Fig. 1.

Mapping neurons’ input–output functions in awake mice using cellular-resolution optogenetic stimulation. (A) Schematic of experiments in V1 of awake mice during i) spontaneous activity and ii) visual stimulus presentation. (B) Schematics of some possible input–output (IO) functions. Optogenetic responses at different points along the IO function can uncover the shape and extent of nonlinearities. (C) Beyond the shape of the IO function, a key unknown feature is where neurons sit on their IO curves during spontaneous activity (schematized by vertical green lines).

The paper proceeds as follows. We deliver visual stimulation to change cells’ activity levels. We first deliver optogenetic stimuli paired with high-contrast visual stimuli that produce a wide range of responses, both suppression and activation, across populations of neurons in V1 (Fig. 2). Using these data, we infer the shape of the average IO function across cells. We then show that this average shape is consistent with that seen in individual cells (Fig. 3). To do this, we first vary visual stimuli to alternately suppress and activate the same cell and then vary optogenetic stimulation intensity. Next, we reconstruct this IO function, inferring cells’ average operating point (Fig. 4; see Fig. 1C for a schematic), and characterize variability of IO functions across cells. We end by discussing the major features of the IO function that we have uncovered.

Fig. 2.

V1 neurons attenuate inputs when suppressed. (A and B) Experiment schematic. Two-photon calcium imaging during visual, cell-specific optogenetic, or paired optogenetic + visual stimulation in awake mice. (C) DIO-jGCaMP8s-P2A-stChrimsonR expressing in V1 layer 2/3. (D) Visually evoked responses to a drifting grating: suppressed (blue) and activated (red) neurons. (E) The schematic showing increases/decreases of optogenetic responses during vision reveals nonlinear amplification/attenuation of inputs. (F) Example visually activated cell. Visual response (Left) and paired Opto + Vis response (Right). (G) Optogenetic response during vision (O + V, Left) and Opto-only response (Right). (H) No change in optogenetic response during and without vision (linear regime). (I–K) Same as F–H for a visually suppressed cell with decreased optogenetic response during vision (nonlinear regime). (L) Trial-averaged (mean ± SEM, N = 50 reps) activity traces for the same visually activated (Top) and visually suppressed (Bottom) example cells in F–K. (M), Comparison of optogenetic responses during and without vision. The green arrow depicts attenuation in visually suppressed cells. (N) Visually activated cells (visual response between 5% and 60% ΔF/F₀; N = 70) show minimal change in optogenetic response (Top). Strongly activated cells (visual response > 60% ΔF/F₀; N = 30) show decreased optogenetic response, suggesting saturation (Middle). Visually suppressed cells (visual response < −5% ΔF/F₀; N = 94) show attenuated optogenetic response (Bottom). *P < 0.05 and ***P < 0.001; Wilcoxon signed-rank test with Bonferroni correction for multiple comparisons. Black lines: mean ± SEM. (O) Change in optogenetic response in neurons (N = 364 total cells, 15 ± 1.3 cells responsive per pattern; N = 25 patterns across N = 7 animals) across the working range of visual responses. Black line: LOESS fit to data with 95% CI via bootstrap.

Fig. 3.

Individual V1 neurons exhibit nonlinear responses when suppressed and linear responses when activated. (A) Experiment schematic showing cells suppressed and activated by orthogonal gratings, moving them down and up along their IO function. (B) Example neuron suppressed and activated by two different visual stimuli (Left). Trial averaged traces (mean ± SEM; N = 50 trials) in Opto-only and Opto + Vis conditions for both visual stimuli. (C) Cell-averaged (mean ± SEM; N = 6) optogenetic responses during and without visual stimuli. (D) Change in optogenetic response plotted against visual response for all cells. Gray lines: individual cells at their suppressed and activated states. Black line: mean across cells. Crosses are SEM of change in optogenetic response and visual response across cells. Blue: suppressed. Red: activated. *P < 0.05; Wilcoxon signed-rank test. (E) Experiment schematic showing cells stimulated with two optogenetic powers (lo: 6.5 mW/target, hi: 8 mW/target). (F) Example neuron stimulated with both optogenetic powers during and without visual stimulation. (G) Cell-averaged (mean ± SEM; N = 27) optogenetic responses in visually suppressed cells for vis-on and vis-off conditions. Inset: As cells are visually suppressed, they move to a lower slope region of their IO function, measured between the two points of optogenetic powers. (H) Same as G, for visually activated cells (N = 25). Inset: As cells are visually activated, they remain in a linear regime of their IO function. (I) Difference in optogenetic response between low and high optogenetic powers. Gray points: individual cells. Median ± SEM across cells plotted for vis-on (blue: suppressed; red: activated) and vis-off (black) conditions. Suppressed cells show decreases in response difference between optogenetic powers. *P < 0.05; Wilcoxon signed-rank test (vis on vs. vis off) or Mann–Whitney U rank test (vis off vs. vis off) with Bonferroni correction for multiple comparisons.

Fig. 4.

V1 pyramidal cells operate at the transition point of a supralinear-to-linear IO function. (A) Change in optogenetic response as a function of visual response. Red line: LOESS fit to cell data (N = 364 cells) with 95% CI. (B) Estimated IO function and 95% CI calculated by shifting and integrating the fit trend line in A (Methods). Green line: the operating point of V1 cells, set as the point in which cells’ visual response is 0% ΔF/F₀ (green line in A). Black line: Ricciardi transfer function fit to the estimated IO function (±95% CIs via bootstrap; Methods). (C) Schematic of the underlying IO function of the average V1 pyramidal cell showing a supralinear regime for low inputs and a linear regime before response saturation for high inputs. Cells operate between the supralinear and linear regimes, allowing cells suppressed below their spontaneous activity levels to attenuate inputs.

Cell-Specific Optogenetic Stimulation during and without Visual Stimulation Reveals Attenuation of Inputs with Suppression.

We expressed the soma-targeted opsin stChrimsonR and the calcium indicator jGCaMP8s via a single Cre-dependent virus (in the Emx1-Cre mouse line) and stimulated V1 excitatory pyramidal cells (Fig. 2A) at cellular resolution while imaging them. To compare responses to optogenetic stimulation at different points along cells’ IO functions, we stimulated during both spontaneous conditions (i.e., no concurrent sensory stimulation), as well as during presentation of a visual stimulus (Fig. 2B; trial types randomly interleaved; Methods). Many cells change their activity in response to drifting gratings, with many increasing their firing rates (visually activated) and some decreasing (visually suppressed; Fig. 2 C and D and SI Appendix, Fig. S1).

We applied optogenetic stimulation while cells’ firing rates were changed during visual stimulation, as well as during periods of spontaneous activity with no visual stimulus (visual stimulus: 3 to 4 s, optogenetic stimulus: 300 ms, with onset 1.5 to 2.0 s after visual stimulation onset; Methods). We compared the response produced by optogenetic stimulation during the visual stimulus (Opto + Vis, Fig. 2F; incremental response during visual stimulation defined as (Opto + Vis)-Vis, Fig. 2G) to the response during optogenetic stimulus alone (Opto, Fig. 2G).

If the IO function follows a power law (is concave up; Fig. 1B), the optogenetic response should increase as neurons’ activity increases (schematic: Fig. 2E). If cells instead operate in a linear regime, they should produce a similar response increment to optogenetic input even as neurons’ activity varies.

For many neurons whose firing rates were increased by visual stimulation, the response to stimulation was similar when optogenetic stimulation was delivered alone or with visual stimulation (example cell, Fig. 2H; population, Fig. 2N). Neurons that were moderately activated by the visual stimulus (N = 70 cells; ΔF/F₀ > 5% and <60%) showed nearly the same response in the two conditions (mean optogenetic response during spontaneous condition: 22.7% ΔF/F₀, paired visual condition: 21.3%; P = 0.02, Wilcoxon signed-rank test with Bonferroni correction for multiple comparisons; Fig. 2 N, Top). This indicates that above neurons’ firing rate at rest, neurons’ average IO function is nearly linear.

In cells exhibiting large increases in firing rate, we found some signs of saturation. Neurons more strongly activated by the visual stimulus (N = 30 cells; ΔF/F₀ > 60%) showed decreased responses to optogenetic input (P = 2 * 10⁻⁴, Wilcoxon signed-rank test with Bonferroni correction; Fig. 2 N, Middle).

In contrast to the linear effects we observed for increases in firing rate, many cells suppressed by visual stimulation showed substantial attenuation of their response to optogenetic input (Fig. 2 I–K). The attenuation was often dramatic, as in the example cell (Fig. 2 L and M, Bottom) whose response was reduced to less than 50% of its response before suppression. Attenuation was widespread across the population of suppressed neurons (N = 94 suppressed cells, ΔF/F₀ < −5%, attenuation average: 47.3%; P = 2 * 10^–16, Wilcoxon signed-rank test with Bonferroni correction; Fig. 2 N, Bottom). These observations suggest that below neurons’ resting activity levels (below spontaneous activity) the IO function shape is not linear, and instead shows a substantial nonlinearity.

The timecourses of neurons’ responses to optogenetic input (Fig. 2 L and M) were as expected for both suppressed and activated neurons, featuring a fast increase in fluorescence during optogenetic stimulation consistent with induced spiking, followed by a slower decay after optogenetic stimulus offset consistent with calcium decay dynamics. Across trials, visual responses remained relatively consistent, with minimal response adaptation. Optogenetic responses slowly decreased across trials, but our results were consistent when analysis was restricted to either the early or the late trials (SI Appendix, Fig. S2).

To characterize neurons’ IO functions across a range of activity levels, we plotted responses across many optogenetic stimulation sessions in many cells, spanning the working range of visual responses—from strongly suppressed to strongly activated (N = 11 experiments in N = 7 mice; N = 364 total cells stimulated). We plotted the average change in optogenetic response as a function of the neurons’ responses to visual input just before receiving optogenetic stimulation (Fig. 2O). As cells were more suppressed, we found, on average, more attenuation (via LOESS regression; Methods). This attenuation was seen across individual animals and experiments (SI Appendix, Fig. S3).

The effects did not depend on viral preparation (bicistronic virus or separate viruses with opsin and GCaMP; SI Appendix, Fig. S4). The effects also did not covary with baseline fluorescence across cells or animals, suggesting that the results are not changed by differences in GCaMP expression, or opsin expression levels, which are related due to the bicistronic viral expression method (39–41) (this analysis performed only on results using bicistronic virus; N = 9 of 11 experiments; SI Appendix, Fig. S4). The effects also are unlikely to be affected by optical cross talk, inadvertent activation of opsin during imaging, as we found little evidence for cross talk with the stChrimsonR opsin (39). This is likely due to its smaller charge transfer per two-photon activation event compared to other commonly used opsins (42).

The results also did not change when the visual stimulus was varied. Neurons showed the same attenuation-by-suppression, with linear responses above spontaneous activity (SI Appendix, Fig. S4), whether animals were presented with full-field gratings (drifting at 2 Hz; Methods) or small oriented patches (Gabors; full width at half max, FWHM = 15 degrees; aligned to cortical retinotopic location, also drifting at 2 Hz).

These results were not due to potential nonlinearities in the calcium indicator. We selected jGCaMP8s for this work because it is nearly linear below saturation, and thus for low spike counts its fluorescence represents average spike numbers without distortion. Still, to ensure these effects were not distorted by indicator properties (i.e., nonlinearities between spikes and fluorescence intensity), we deconvolved fluorescence traces using methods that account for indicator nonlinearities. The attenuation seen for suppression and the largely linear behavior for elevated rates (Fig. 2O) remains present in deconvolved data (SI Appendix, Fig. S5), including for deconvolution methods that do (43) and do not (44) account for indicator onset nonlinearities, as expected from the nearly linear properties of jGCaMP8s for low spike rates. Because deconvolution can be sensitive to estimation convergence and data quality, we further confirmed these results by recreating fluorescence traces using a spike-to-fluorescence model (45) that either did or did not include indicator nonlinearities (SI Appendix, Fig. S6). All these analyses support the idea that our measurements are not significantly distorted by the calcium indicator.

The number of cells we stimulated, kept to small groups of cells at maximum, did not affect our results. In these optogenetic experiments (Fig. 2), we targeted small groups of cells for stimulation (5 to 12 cells) to increase yield by stimulating multiple neurons in parallel while avoiding triggering larger network interactions, which have been reported to occur at and above approximately twenty stimulation targets (37–39, 46). Stimulation of small groups like the ones we use has been found to induce only small network effects – typically weak, broad suppression (47–49) (refs. 46 and 50 for simulations across input pattern size). The radial extent of our stimulation targets is limited in the XY plane (radial optical extent, full width at half max, FWHM = 9.4 µm), and we have previously obtained evidence that activation with these procedures is generally restricted to single cells in XY (39). However, we note it is possible that our stimulation disks could in principle activate occasional off-target cells directly above or below the focal plane (measured axial FWHM = 54 µm). To confirm that our responses are not affected by the number of cells stimulated simultaneously, we changed the number of stimulated cells, performing experiments with only one stimulation target at a time (SI Appendix, Fig. S7), and found that the results were consistent with our multitarget stimulation experiments.

Together, these results imply the average IO function of pyramidal cells in mouse V1 exhibits a linear regime above, and a nonlinear regime below, the resting activity level of neurons in the awake state. The nonlinearity below rest attenuates inputs so that suppression leads to attenuation.

Consistent Effects—Attenuation, Not Amplification—Measured in Individual Pyramidal Cells.

We next asked whether responses reflected by the cell-average trends were present within individual neurons. First, we identified cells whose activity was suppressed by a drifting grating of one direction but increased by the orthogonal direction (Fig. 3A). We then optogenetically stimulated these cells as they fired spontaneously and also while we presented either grating (Fig. 3 B and C; N = 6 cells in N = 2 animals). We found results for single cells that are consistent with our previous results from populations of cells: suppression led to attenuation of responses, while optogenetic stimulation applied when cells’ activity was elevated by visual input produced similar responses (i.e., a linear response; P = 0.031; Wilcoxon signed-rank test; Fig. 3D).

We next examined whether stimulation of a cell with multiple optogenetic intensities (Fig. 3 E and F) was consistent with the results we found in averages over cells (Fig. 2). If neurons have a nonlinear region below their resting point, then we should expect a smaller difference between responses (smaller slope) to the two stimulation intensities in suppressed cells. Consistent with this idea, we found that as cells were suppressed, the difference in responses between high and low powers decreased (Fig. 3G, N = 27 cells in N = 3 animals; Fig. 3 I, Left, P = 0.027; Wilcoxon signed-rank test with Bonferroni correction). Similarly, a linear region above cells’ resting point implies there should be little or no difference in responses as cells’ firing rates increase. Confirming this, we found no difference in responses at either optogenetic power (N = 25 cells in N = 3 animals; Fig. 3H) and found there was no difference (change in slope) between high and low powers (Fig. 3 I, Right and SI Appendix, Fig. S8).

In sum, these results confirm that individual neurons show IO functions that are consistent with the population data in Fig. 2. These IO functions are nonlinear below resting activity levels, showing attenuation-by-suppression, and linear for moderate increases in firing rate—across much of the range evoked by visual input.

Cells Operate at the Transition Point of a Supralinear-to-Linear IO Function at Spontaneous Activity Levels.

We next sought to estimate the IO function across the range of activity levels we measured. To do this, we used changes in optogenetic responses (Fig. 2K) to infer the slope of the underlying function. We integrated the curve fitted across all cells’ change in responses (Fig. 2O) to generate the underlying IO function (Fig. 4 A and B). Consistent with our individual observations, this IO function has a supralinear region followed by a linear region, before eventual saturation at high inputs (Fig. 4B).

One constraint with our approach is that we obtain the IO function by integrating changes in optogenetic response as a function of changes in ongoing neural activity. This method characterizes changes in response, but does not define zero firing rate (X-axis, Fig. 4B). To ensure that the shape of the IO function we found is valid even at low activity levels, we characterized the impact of interpolated extensions to the fitted line on the resulting IO function. We found no qualitative differences (SI Appendix, Fig. S9) from our prior estimates. To estimate the zero point more directly, we estimated neurons’ firing rate based on their calcium transient rate by deconvolving each neuron’s response. This analysis revealed that there are indeed cells whose activity reaches zero, approximated by a zero rate of calcium transients, during visual stimulation (SI Appendix, Fig. S9). We used this value to represent zero firing rates in Fig. 4B, and also found that deconvolved responses yield the same shape of the underlying IO function as when using calcium responses (SI Appendix, Fig. S10).

The IO function shape we found is comparable to the Ricciardi function, an IO function that arises in spiking network models that are recurrently connected. Its shape is influenced by the inputs neurons receive from other neurons in the network, because variability in membrane potential and membrane conductance influence IO functions (8), and these vary with input. The Ricciardi function shows a supralinear, then a linear, regime with large or small amounts of saturation depending on input and other parameters such as maximal firing rate and refractory periods (29). We fit the parameters of the Ricciardi function to our data (Methods), finding a strong similarity between the model and our data (Fig. 4B). Compared to other models of the IO function, like threshold-linear or power-law functions, the Ricciardi function produces the closest fit to our data (SI Appendix, Fig. S11). While other functions could also be used to describe our data, it is notable that the Ricciardi provides a good fit, as it arises as the analytical form of the IO function of a neuron embedded in a recurrent spiking network model.

To assess the variability of the IO function fit across neurons, we calculated bootstrapped CI of the Ricciardi function fit (Fig. 4B; on each bootstrap trial, fitting a LOESS, integrating, and fitting the Ricciardi function to the resulting IO function estimate; parameter sensitivity analysis in SI Appendix, Fig. S12).

Our results are summarized in Fig. 4C. Neurons operate during spontaneous activity near the transition from supralinear to linear regions of their IO function. Suppression then moves neurons into the supralinear region so that the more a neuron is suppressed, the smaller its response to a fixed input. Thus, the nonlinearity in the IO function, and the operating point of the neurons during spontaneous activity, together produce attenuation-by-suppression.

Trial-to-Trial Response Variability Reveals Differences in IO Function Gain, but Not IO Function Shape.

The supralinear-to-linear IO function shape we find is an average across observations from many neurons. To explore variability that might exist across cells, we leveraged the trial-to-trial fluctuations in cells’ activity to examine responses at multiple points along a cell’s IO function (Fig. 5A). From one trial to the next, neurons’ activity (ΔF/F₀) fluctuates. Therefore, on different trials, cells may be more or less excited (or more or less suppressed) just before the onset of optogenetic stimulation. By relating the magnitude of neurons’ optogenetic responses to the prior ΔF/F₀ levels, we obtain multiple measurements along the IO function for each cell, allowing us to examine the shapes of individual cells’ IO functions.

Fig. 5.

Trial-to-trial variability reveals that IO function shape is consistent across cells, while IO function gain is variable. (A) Schematic of analysis: There is trial-to-trial variability prior to optogenetic stimulation that causes each trial’s optogenetic stimulus to arrive at a different baseline level of activity. (B) Example cells. X-axis: activity prior to optogenetic stimulation. Y-axis: optogenetic response. (Left) suppressed neuron, (Right) activated neuron. Colored lines: individual fits across trial data. Insets: schematic of the IO function segment that each fit line represents, computed by integrating the fit. (C) Fit slopes across all suppressed (Left, blue: visual response < −5% ΔF/F₀; mean ± SEM = 0.29 ± 0.04; N = 82) and activated (Right, red: visual response > 5% ΔF/F₀; mean ± SEM = −0.04 ± 0.01; N = 96; Right) cells. This parameter gives the curvature of cells’ IO functions. Dashed colored lines: mean slope. ***P < 0.001, Wilcoxon signed-rank test with Bonferroni correction. (D) Same as C, but for fit intercepts across cells, which give the gain or slope of cells’ IO functions. Gray histograms: variability of fit intercepts across cells. (E) IO function segments derived by integrating fits in B; all lines normalized to the single largest peak response.

To characterize cells’ IO function shapes, we first estimated the change in optogenetic response as a function of preceding activity. We used a mixed-effects model,

$$\begin{array}{c}{\text{opto}}_{\mathrm{i},\mathrm{j}}={\beta }_{0,\mathrm{i}}+{\beta }_{1,\mathrm{i}}(\text{preceding}\mathrm{\Delta }\mathrm{F}/{\mathrm{F}}_0{)}_{\mathrm{i},\mathrm{j}}+{\beta }_2\\ + {\beta }_3 (\text{baseline} \text{fluorescence}{)}_i\end{array}$$ [1]

to estimate effects of preceding activity while accounting for cells’ baseline fluorescence levels (β₀, β₁ random effects, other coefficients fixed effects, i indexes cells, j trials; Methods). For cells transfected with the bicistronic virus that express both opsin and indicator, the baseline fluorescence is a proxy for opsin level (same cells as in Fig. 2N, data from bicistronic virus expression animals only, N = 6 animals, N = 9 experiments; N = 178 total cells). We pooled our data across trials with only optogenetic input, and with both optogenetic and visual (O + V) input. This is because as expected for trials with larger mean responses due to visual input, the variability on O + V trials was larger, allowing exploration of a larger portion of the IO function. Few trials overall for a given suppressed (or activated) cell crossed above (or below) the mean spontaneous activity level (0% ΔF/F₀). Therefore, we used only trials where the activity before optogenetic stimulation was either below (suppressed; range 55 to 81 trials) or above (activated; 44 to 82 trials) spontaneous activity. This limits distortion of fit slopes that could result from fitting one curve across both the supralinear and linear regimes (Fig. 3 A–D).

This model yields a fit line for each cell (Fig. 5B). This fitted line represents the derivative of the cell’s IO function so that the slope of the line indicates the curvature of the IO function. That is, a slope near 0 reflects linearity, a slope > 0 reflects supralinearity, and a slope < 0 reflects sublinearity. The intercept of the fitted line corresponds to the slope, or the gain, of the IO function at cells’ resting point (i.e., where the preceding ΔF/F₀ = 0%, or, where there is no change from baseline activity).

Examining the distribution of fit slopes across all visually activated and suppressed cells reveals IO function shapes consistent with our previous findings. First, we found visually suppressed cells primarily operate along a supralinearity in the IO function (Fig. 5 C, Left; N = 82 cells; mean fit slope = 0.29 ± 0.04; P = 2 * 10^–11, Wilcoxon signed-rank test with Bonferroni correction). On the other hand, cells whose activity was elevated above resting levels show near-linear IO functions, though many operate along slightly sublinear IO functions, suggesting a transition from linear to saturating responses (Fig. 5 C, Right; N = 96 cells; mean fit slope = −0.04 ± 0.01; P = 2 * 10⁻⁷, Wilcoxon signed-rank test with Bonferroni correction).

Next, we used the distributions of fit intercepts to determine whether gain, or IO function slope, varies across cells (Fig. 5D). This measurement is enabled by the use of trial-to-trial variation. The result extends the analyses in Fig. 2, which average across cells when computing slopes. The principal result is that while excited cells have similar near-linear shapes, their gain (slope) varies from cell to cell. Note that this variation in gain could be due to differences in opsin expression, or due to underlying variation in cells’ gain, perhaps due to receiving different levels of network input (8). For both suppressed and activated cells, we found significant variance (P < 0.001 for both groups, via permutation tests; SI Appendix, Fig. S13). We integrated cells’ fits to construct the segment of their IO function corresponding to the fits (Fig. 5E; qualitative comparisons of IO function shape across neurons in SI Appendix, Fig. S13). Remarkably, cells are quite consistent in shape within groups. Suppressed cells show qualitatively similar supralinear shapes. Activated cells are linear or near-linear, with moderate saturation at high levels of activity, but their gain varies significantly (Fig. 5D).

In sum, we find that above rest or spontaneous activity levels, cells consistently operate in a near-linear regime with some saturation, and below rest, show a supralinear IO function shape. Despite the variability we measured in the overall slope, or gain, of the IO functions, the attenuation-by-suppression effects are consistent across cells.

Discussion

Here, we demonstrate through direct stimulation that excitatory cells in the cortex of awake mice operate along a supralinear-to-linear IO function. Cells’ f-I curves, or IO functions, have previously been suggested to exhibit smooth nonlinearities, such as power-law nonlinearities, across the working range of visual responses (14, 23, 25), and it was thought these nonlinearities would serve to amplify inputs (51, 52). However, we find a key feature of V1 excitatory cells is not that they exhibit amplification above baseline (spontaneous) activity levels, but rather, that they show linear responses. We do find that neurons’ IO functions show a smooth nonlinearity—a supralinearity—but this effect occurs below the level of spontaneous firing on average. Thus, we conclude that the prominent nonlinearity in neurons’ IO functions does not allow amplification, but instead causes neurons to attenuate their responses to input as they are suppressed.

Our inferred IO function shape (Fig. 4B) has a supralinear region, which we show here is below resting or spontaneous activity, a linear region that covers most of neurons’ operating range as defined by high visual stimulus contrast (100% contrast, Figs. 2 and 3 A–D), and a weakly sublinear region for the highest activity levels.

Relationship to Prior Work.

Here, we measure the somatic IO function, or how neurons’ spike rates change as a function of aggregate inputs from the dendrites. While dendritic nonlinearities may change how certain inputs are integrated, the somatic IO function is a good description of neurons’ full IO function when inputs are summed linearly in the dendrites. While nearly linear summation in dendrites can occur in some conditions (53, 54), nonlinear processes in the dendrites can add to the neural computations performed by recurrent networks of neurons (e.g., refs. 55–57). Understanding brain function will require understanding both the recurrent network computations that occur via near-linear dendritic summation as well as the computations created by dendritic nonlinearities.

Two prior observations lend credence to our findings. First, our data are strikingly similar to the Ricciardi IO function that is derived from integrate-and-fire model networks (28, 29). Ricciardi functions can have dramatic saturation at high activity levels, and thus a major result of our work is that the normal operating range of cortical neurons is largely below saturation, in a near-linear regime for elevated rates. Previously, it was unknown whether the Ricciardi function was a good description of cortical neurons’ responses in vivo. Second, a set of dynamic clamp experiments which simulated conductance inputs finds IO function shapes that are also similar to our results (33).

Prior studies have estimated a power law relationship between membrane potential and spike rate in vivo, with an expansive supralinearity throughout neurons’ input range (23, 25, 31). While the shape of a power law in some regimes can approximate the supralinear-to-linear IO function we see, prior characterizations of a power law in the relationship between membrane potential and spike rate do not necessarily mean the IO function follows a power law. This is especially true at higher spike rates, when neural firing rates become more dependent on refractory periods and time to integrate to threshold (29), instead of spikes being governed by membrane potential fluctuations (58). Also, direct input injection, as we use, is better able to determine where the supralinearity in the IO function is relative to spontaneous activity. This allowed us to determine that neurons sit near this transition during spontaneous firing. In addition, our all-optical approach has allowed us to measure the responses of a large population of neurons, and these data suggest that while individual neurons share the same supralinear-to-linear IO shape, there is variability in gain or slope across cells (Figs. 4 and 5). Such variability across cells may serve to improve information coding at the population level, as has been suggested of the variability found in neurons’ tuning curves (59).

Our results might have, in principle, been changed by off-target or network responses driven in other neurons by our stimulation. Yet individual cortical synapses are relatively weak, and stimulation of more than a few cells is generally needed to see substantial responses in nontargeted neurons (37–39). Consistent with that idea, stimulating with one target at a time does not change our results (SI Appendix, Fig. S7), suggesting that any potential activation of other neurons through network effects does not affect our conclusions.

We observed some saturation in IO functions for neurons that showed the largest responses to strong input. This effect could be caused by mechanisms such as sodium channel inactivation in single cells, strong inhibitory coupling in the network, and/or consequences of the refractory period (13, 29). However, we cannot rule out that decreased responses were due to saturation of fluorescent signals at high activity levels (45).

Recurrent Network Input Likely Plays a Role in Creating the Supralinearity.

What shapes the supralinearity we measure for low inputs? In principle, this soft threshold for the somatic IO function could arise from biophysical factors like voltage-gated channels. If that were the case, we might expect to see the supralinearity in in vitro measurements of f-I curves, but these often show a threshold-linear response function, not a soft threshold or supralinearity (2). Because in vitro networks in slices or culture are often in a lower-activity state than the network is in vivo, these observations suggest the supralinearity we observe may arise from inputs to cells from other neurons in the network. Neurons in the cortex receive a constant barrage of excitatory and inhibitory inputs, which can lead to ongoing fluctuations in their membrane potentials (60–63). These fluctuations can smooth the relationship between input and output near the IO function threshold (9, 10). In fact, the Ricciardi IO function that matches our data is an analytical description of the IO function shape produced from inputs in a recurrent network of excitatory and inhibitory neurons (28, 29).

Implications for Network Computation and Brain Function.

Two major consequences for brain function arise from these results. First, the attenuation-by-suppression we observe can have direct consequences on cortical computation—in other words, the brain can filter out inputs to neurons in certain contexts, by suppressing the activity of those neurons so their incremental responses to other inputs are attenuated. Second, and more speculatively, because the supralinear-to-linear somatic IO function we characterized shares general features with activation functions used in modern machine learning systems, including generative AI systems like ChatGPT (64), our results may have implications for learning.

Attenuation-by-suppression has computational consequences for brain operation. Neurons are suppressed in a wide variety of sensory contexts. A classic example is the case of sensory surround suppression. In surround suppression (65–67), sensory stimuli outside a neuron’s key feature tuning (outside its receptive field) decrease that neuron’s firing. Surround suppression, combined with the attenuation-by-suppression effect we observe, could lead to sharpening of responses to small visual stimuli or stimuli that change quickly in space, like edges. In this scenario, as some cells are activated by the stimulus, those activating inputs, falling on neurons’ linear region, would be passed on to downstream areas. In contrast, other nearby neurons will have receptive fields that are in the surround—the small stimulus would fall outside their receptive field, causing them to be suppressed (17, 68–71). These suppressed neurons would then attenuate any additional inputs that arrive, filtering out those inputs, so that surround inputs have less impact on downstream neurons. This effect could, at the population level, serve to amplify relevant responses above background noise (72).

Notably, this mechanism is not just the usual conception of suppressive surrounds. Here, we show not just that visual stimuli suppress some neurons’ activity, but that suppressed neurons are actively filtering out the inputs they receive—that is, attenuating added inputs, or attenuation-by-suppression. We arrive at this result by going beyond the measurements possible with visual stimuli alone, using an in vivo cell-specific stimulation approach that allows us to directly measure IO functions and changes in response to fixed input.

Attenuation-by-suppression is related to the idea that expansive nonlinearities due to spike threshold can sharpen responses (24, 73); this type of sharpening due to the IO function may be more common across cortical areas than has been previously reported. Attenuation-by-suppression may also offer an explanation for perceptual masking effects arising from optogenetic stimulation (74), as optogenetic stimulation can lead to suppression caused by recurrent excitatory connections (38, 46). These effects, where optogenetic stimulation generates suppression and thus attenuation, may also underlie the differences in visual perception noted when visual and optogenetic inputs are combined (75, 76).

Another major result of this work is that our measured IO functions share important features with the activation functions used in recent AI systems. Over a decade ago, activation functions with a linear regime over much of the unit’s activity range (rectified-linear units, ReLUs) were shown to yield some learning improvements compared to sigmoidal functions. Since then, it has been observed that learning performance can be additionally improved by activation functions that smooth the ReLU’s sharp transition from zero activation to the linear region (77, 78). Intuitively, this smoothing avoids a zero gradient below the rectification threshold and limits the degree to which synapses drop out of learning due to a lack of available error signal. One might map this to biology by saying that neurons that “cannot fire never wire.” Thus, while learning rules in vivo are not fully understood, if there were a sharp IO function threshold in vivo, it might be that learning would be difficult. This smooth nonlinearity followed by a linear region is a shape shared by many activation functions in modern AI systems, including the softplus, Gaussian error linear (GELU), and exponential linear (ELU) functions. Our data show striking similarities to these activation functions, with a smooth transition around the nonlinearity and a largely linear regime through most of the range used for even high-contrast visual responses. Since these activation functions work well in model networks, we might speculate that biological neurons have faced selection pressure to produce IO functions with this shape, to optimize some form of learning or other computation.

Conclusion.

In summary, our findings characterize the average IO function of neurons in the cortex during normal brain operation—in the awake state, as animals actively receive sensory input. The attenuation-by-suppression we observe suggests a key role for ongoing activity in cortical networks. Because ongoing input creates the smooth nonlinearity below spontaneous firing rates, it can perform computations by allowing neurons to selectively filter out extraneous features of sensory input.

Methods

Viral Injections and Cranial Window Implants.

All procedures were approved by the NIMH Institutional Animal Care and Use Committee (IACUC) and conform to relevant regulatory standards. Emx1-Cre mice (The Jackson Laboratory) were used in all experiments to target expression of Cre to excitatory neurons. Adult mice were anesthetized with isoflurane (1 to 3% in 100% O₂ at 1 L/min) and administered intraperitoneal dexamethasone (3.2 mg/kg). A custom metal head post was fixated at the base of the skull. A 3 mm diameter circular craniotomy was made over the left hemisphere of the primary visual cortex (ML −3.1 mm, AP +1.5 mm relative to Lambda).

A bicistronic virus expressing both opsin and GCaMP indicator, AAV9-hSyn-DIO-jGCaMP8s-P2A-stChrimsonR (39) (Addgene, 174007) was diluted in phosphate-buffered saline (final titer: 2.6 * 10¹² GC/mL, N = 3 mice; 2.9 * 10¹² GC/mL, N = 3 mouse; 3.4 * 10¹² GC/mL, N = 2 mice; 4.7 * 10¹² GC/mL, N = 2 mice; 6.5 * 10¹² GC/mL, N = 2 mice) and injected 200 µm below the surface of the brain (multiple injections, 300 nL/injection, 0.1 µL/min) to target expression to layer 2/3 neurons. In N = 1 mouse, separate viruses of AAV9-hSyn-jGCaMP8s-WPRE (1.0 * 10¹³ GC/mL) and AAV9-hSyn-DIO-stChrimsonR-mRuby2 (2.7 * 10¹² GC/mL) were instead injected.

A 3 mm optical window (Tower Optical) was implanted over the craniotomy. Both the optical window and metal head post were fixed to the skull using C&B metabond (Parkell) cement dyed black. Animals were individually housed after surgery. Mice were imaged three or more weeks postinjection and water-scheduled (1.0 mL/d) to promote alertness during experiments that used intermittent water delivery.

Two-Photon Calcium Imaging.

Two-photon calcium imaging was performed with a custom-built microscope (MIMMS, Modular In vivo Multiphoton Microscopy System, components, Sutter Instruments; controlled by ScanImage, MBF Biosciences; bidirectional 8 kHz resonant scanning, 512 lines, ~30 Hz frame rate). Calcium responses were imaged with 920 nm laser pulses (Chameleon Discovery NX laser, Coherent, Inc., pulse rate 80 MHz, pulse energy 0.19 to 0.25 nJ/pulse; imaging power 15 to 30 mW measured at the front aperture of the objective; 16× objective, 0.8 NA, Nikon Inc; immersion with clear ultrasound gel, ~1 mL; 100 to 200 µm below pia, L2/3 of V1, field of view 400 × 400 µm).

Cell-Specific Two-Photon Stimulation.

Cell-specific photostimulation was performed using either 1,030 nm (Satsuma laser, Amplitude Laser; Figs. 2 and 3 E–I) or 1,035 nm light (Monaco laser, Coherent, Inc.; Fig. 3 A–D). The laser wavefront was shaped by a spatial light modulator (1,920 × 1,152 pixels; Meadowlark) to target patterns of cells for stimulation (10 µm diameter disks; measured size is 9.4 µm radial and 54 µm axial full-width at half-max; 300 ms, 8 mW/target, 6.5 mW/target for low power in Fig. 3 E–I, 500 kHz pulse rate; TTL logic gate, Pulse Research Lab, used to invert imaging line clock to gate laser during pixel acquisition and stimulate during reversal of bidirectional scanning: on time 19 µs, off time 44 µs, 30% duty cycle; stimulation power accounting for blanking: 2.4 mW/target, or 2 mW/target for low power; ref. 39 for more details.

Visual Stimulation.

Visual responses were measured in awake, head-fixed mice viewing sinusoidal drifting grating stimuli (full-field or Gabor patches filtered with a 15° full-width half-max Gaussian mask; spatial frequency of 0.1 cycles/degree; speed 20 deg per sec; drifting at 2 Hz; 100% contrast; 3 to 4 s; stimuli presented on an LCD monitor with neutral gray background).

To align Gabor patches to the receptive fields of neurons in the imaging FOV, we performed retinotopy via hemodynamic intrinsic imaging using 530 nm light (delivered via fiber-coupled LED, Thorlabs; 2 Hz imaging) (79). Retinotopic maps were calculated by measuring hemodynamic-related changes in light absorption across the cortex to square wave drifting gratings (0.1 cycles/degree, 10° diameter, 6 locations, 5 s; 5-s baseline window and 2.5-s response window starting 3 s after stimulus onset used to measure responses).

Experimental Sequence.

To ensure that mice were awake and alert, small volumes of water were delivered (3 µL per trial, ~300 µL total per session; 20% of trials with independent probability; 50 to 60-s average time between delivery; water delivered 2.5 to 3.5 s before onset of optogenetic stimulus and 2 s before onset of visual stimulus).

For all experiments, visual responses were measured to drifting grating stimuli of different directions (8 directions 45° apart randomly interleaved, 20 repetitions per direction, 2 s on, 6 s off). Mice were head-fixed and sat in a tube during experiments. For optogenetic stimulation experiments, a single direction was chosen that led to activation or suppression of a number of neurons in the field of view (2 to 3 patterns of neurons per experiment; Figs. 2 and 3 E–I). During these experiments, trials with visual only, optogenetic only, and paired stimulation trials were randomly interleaved; visual stimulus always the selected single drifting grating; 50 repetitions per trial type per optogenetic stimulation pattern, 250 to 350 trials total; 6 to 9-s interstimulus interval for optogenetic stimulation. Experiments lasted 100 to 120 min total (20 min visual stimulation of multiple directions, 30 min for visual response characterization and neuron selection, 50 to 70 min for optogenetic and visual stimulation experiments).

For experiments in Fig. 3 A–D, two orthogonal visual stimulus directions were chosen. The experimental sequence was the same. We collected visual response data as described above, multiplied average pixel-wise responses for all orthogonal pairs of directions, and selected neurons for stimulation whose value after multiplication was negative, indicating a positive response for one direction (visually activated) and a negative response for the orthogonal direction (visually suppressed).

Analysis of Two-Photon Imaging Data.

Images acquired via two-photon imaging were first rescaled from 512 × 512 pixels to 256 × 256 to ease data processing. Background correction was done by subtracting the minimum pixel value of the average intensity image and setting all remaining negative pixels (due to noise) to zero. We used the CaImAn toolbox (80) for motion correction and Suite2p (81) for cell segmentation to allow manual selection of cell masks.

We measured neurons’ activity to both visual and optogenetic stimuli (ΔF/F₀; F: average intensity across all pixels within a cell’s segmented mask; F₀: average fluorescence across the 45 imaging timepoints, or 1.5 s, just prior to stimulus presentation; F₀ for Fig. 5: for trials where only the optogenetic stimulus was presented, average fluorescence across 45 imaging timepoints, 1.5 s, just prior to the “preceding activity” window). “Opto” trials or “Vis” trials are where the corresponding stimulus was delivered by itself. “Opto + Vis” trials, (O + V), are when visual stimulus and optogenetic stimulus are delivered on the same trial. The “Opto during vis” response, (O + V) − V, Fig. 2M, was found by subtracting the average Vis trace from the average Opto + Vis trace. The deconvolved activity was estimated using the constrained OASIS method (44) or MLspike (tau = 0.4 s; amplitude parameter for each cell defined as the smallest peak of the denoised calcium trace from the OASIS method; ref. 43).

For response ΔF/F₀ maps (Figs. 2 D and F–K and 3B), we measured trial-averaged activity at every pixel (ΔF/F₀; F: average frame across a time window, Fig. 2D, entire visual stimulus period, Figs. 2 F–I and 3B, entire visual stimulus period except the first 1 s, to exclude the initial transient response at stimulus onset, Fig. 2 G–J, 1-s period following optogenetic stimulus onset; F₀: average frame from 1.5 s before onset of the stimulus).

Analysis of Stimulus-Evoked Responses.

Cell responses (points in Figs. 2 N–O and 3 C, D, and G–I) were computed by averaging timecourses in 0.33-s windows (10 frames), just before (baseline) or after (response) optogenetic stimulus onset. “Vis resp” (X-axis, Figs. 2O, 3D, and 4A): response on Opto + Vis trials, in the window just before optogenetic stimulus onset (when only the visual stimulus was present). “Opto” (Fig. 2N): response on Opto-only trials, in response period after optogenetic stimulus onset, minus baseline period just before optogenetic stimulus onset. “Opto during vis,” “(O + V) − V” (Fig. 2N): response period on Opto + Vis trials minus response period on Vis-only trials, with baseline period difference on the same trials subtracted. Change in optogenetic response between visual and spontaneous conditions (Y-axis, Figs. 2O, 3D, and 4A): difference between “Opto during vis” and the “Opto” responses.

Stimulated cells during our experiments were defined as those whose centroid fell within a 15-micron radius of any stimulation target’s center (5 to 12 targets per pattern, average 9 ± 0.5) and whose “Opto” response value exceeded a 5% ΔF/F₀ threshold (N = 15 ± 1.3 stimulated cells per pattern).

IO Function Estimation.

To estimate the underlying IO function of V1 pyramidal cells, output = ƒ(input), we first performed locally estimated scatterplot smoothing or LOESS (82). We fit LOESS curves (rotated axis to maximize variance on the X-axis, using the loess Python package (83): pypi.org/project/loess/) to the data (Figs. 2O and 4A; CI via bootstrap; N = 1,000 repetitions, N = 364 samples).

The resulting fit represents a change in output versus the input level of a cell, or the derivative of the IO function, shifted vertically by a constant:

$$f\left(\mathrm{input}\right)=\int \left[f^{\text{'}}\left(\mathrm{input}\right)+{\mathrm{y}}_{\mathrm{shift}}\right].$$

The value of y_shift is dependent on two factors (SI Appendix, Fig. S9A): any difference between maximum visual suppression and zero firing rate (y_supp,0) and the size of the optogenetic response (y_opto): y_shift = y_supp,0 + y_opto. Deconvolved spiking estimates of calcium activity (SI Appendix, Fig. S9 B–E) suggest that the most-suppressed cells did indeed reach zero response, so we estimate the shift as the minimum of the fit line: y_supp,0 = −min(ƒ’(input)). Assuming that suppression to zero completely attenuates small optogenetic inputs (as expected for an activation function that goes to zero), we set y_opto to zero. While we cannot directly measure this value because any optogenetic activation moves responses away from zero, we confirm that our conclusions are not changed by various extrapolations from our observed data points (SI Appendix, Fig. S9 F–H).

To fit the Ricciardi transfer function to our estimated IO function (Fig. 4B), we used least squares regression. Initial values for the parameters were taken from the study by Sanzeni et al. (29): ${\tau }_i=20\mathrm{ms}$, ${\tau }_{\mathit{rp}}=2\mathrm{ms}$, $V_r = 10 \mathrm{mV}$, $\theta = 20 \mathrm{mV}$, and $\sigma = 10 \mathrm{mV}$. The fitted Ricciardi function in Fig. 4B and 95% CI were generated by taking the mean of the estimated IO functions computed at each bootstrap repetition of the LOESS fits on the data.

For comparison of the Ricciardi transfer function fit with other model IO functions (SI Appendix, Fig. S11), we fit the following equations to our data, where the parameters a, b, and c were fit for each function:

Threshold-linear (rectified-linear, or ReLU):

$$\begin{array}{cc}y=a(x-b),\hfill & x>b\\ 0,& x<=b,\end{array}$$

Power law:

$$y=a{x}^b,$$

Logistic:

$$y=\frac{a}{e^{-b(x+c)}},$$

Stretched exponential:

$$y=a(1-e^{-b{x}^c}).$$

Analysis of IO Variability.

In order to generate fits across trial-to-trial data for individual cells, we fit a mixed-effects model with a fixed effect for baseline fluorescence [a proxy for opsin levels in neurons expressing the bicistronic construct, LaFosse et al. (39)], to control for differences in optogenetic response due to differences in opsin expression. We thus only included cell data from animals expressing the bicistronic construct in this variability analysis. Cell variation in slope and intercept were random effects. “Opto” and “preceding activity” values were calculated as the mean activity in a 333 ms (10 imaging frames) window just after and before optogenetic stimulation, respectively.

Example fits, as in Fig. 5B, are computed using all the terms in Eq. 1. Analysis of variability across cells (Fig. 5 C and D and SI Appendix, Fig. S13) uses cells’ slopes (β_{1, i}), and intercepts [β_{0, i} + β₂ + (baseline fluorescence) * β₃] for each cell.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

Raw data of cell traces and code to generate figure panels are available on Zenodo (84).