Experimental setup and neuronal response.

Over three sessions, mice are passively shown a battery of visual stimuli, consisting of gratings, sparse noise, images, and movies (Fig 1a). The neuronal responses in the visual cortex to said stimuli are recorded using in vivo 2-photon calcium imaging. We focus on the neuronal responses of one particular stimuli, “Natural Movie One”, consisting of a 30-second natural image scene that is repeated 10 times in each of the three sessions. We analyze data from mice belonging to Cre lines with an excitatory target cell class that are imaged in the primary visual cortex. Crucially, a subset of the neurons imaged across the three sessions can be identified, allowing us to study how the neuronal response of said neurons changes across time. The time difference between the three sessions differs for each mouse and is at least one day.

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Fig 1. Setup of passive data and visualization of feature space representations.

(a) Summary of passive data experiment. (b) Summary of response vector extraction across three imaging sessions. [c-f] Visualization of feature space representations. (c) Drift of response vectors belonging to two separate stimulus groups between two sessions. For example, stimulus group 1 might corresponding to the response vectors of the 0 to 1 second time-block of Natural Movie One and stimulus group 2 the 1 to 2 second time-block. (d) For each stimulus group in each session, we perform PCA to characterize the group’s variation. An important quantity is also the group’s mean response vector. (e) Moving to the respective stimulus group’s PC basis, there is a strong correlation between the variance and mean value along a given direction, familiar from Poisson-like distributions. (f) The aforementioned feature space characterization is used to quantify drift. We define the drift vector, d, of a given stimulus group as pointing from the mean response vector of an earlier session (e.g. Session 1) to the mean response vector of a later session (e.g. Session 2). Δt is the time difference between the sessions.

https://doi.org/10.1371/journal.pcbi.1010716.g001

To quantify the neuronal response, we divide Natural Movie One into 30 non-overlapping 1-second blocks. We define the n-dimensional response vector characterising the neuronal response to a given block as the time-average dF/F value over the 1-second block for each of the n neurons (Fig 1b, see Methods for additional details) [11]. Additionally, we define the response vector to only contain neurons that are identified in all three sessions. This will result in a conservative estimate of the amount of drift, as a cell which does not have any activity in an entire session can be missed by the segmentation. Throughout this work, we define the collection of response vectors corresponding to the same stimulus in a given session as the stimulus group, or just group for brevity. Thus, for the passive data, the response vectors of all 10 repetitions of a given time-block in a given session are members of the same stimulus group.

Between sessions, we will see the response vectors collectively drift (Fig 1c). To understand the geometry behind representational drift, we first quantify the feature space representations of the various stimulus groups in each session. The following quantification will be used on the passive data and throughout the rest of this work.

Feature space geometry.

An important quantity for each stimulus group will be its mean response vector, defined as the mean over all m members of a stimulus group in a given session (Fig 1d). Since there are 10 repetitions of the movie in each session, m = 10 for each stimulus group of the passive data. To characterize the distribution around this mean, for each stimulus group in each session, we perform principal component analysis (PCA) over all m response vectors. This gives us the PC_i directions, each of which we can associate with a ratio of variance explained, 0 ≤ v_i ≤ 1, for i = 1, …, N and N ≡ min(m, n) the number of PCs for the particular stimulus group and session. The PC directions are ordered such that v_i ≥ v_j for i < j. We define the dimension of the feature space representation, D, by calculating the “participation ratio” of the resulting PCA variance explained vector, (1) where 1 ≤ D ≤ N. D thus quantifies roughly how many PC dimensions are needed to contain the majority of the stimulus group variance (Fig 1d). We define the variational space of a given stimulus group as the ⌈D⌉-dimensional subspace spanned by the first ⌈D⌉ PC vectors (where ⌈⋅⌉ is the ceiling function). Lastly, to eliminate the directional ambiguity of PC directions, we define a given PC direction such that the stimulus group has a positive mean along said direction.

Across all datasets analyzed in this work, we find the mean along a given PC direction is strongly correlated with its percentage of variance explained (Fig 1e). That is, directions which vary a lot tend to have larger mean values, familiar from Poisson-like distributions and consistent with previous results [34]. Below we will show how the above feature space representations allows us to quantify certain characteristics of drift between sessions (Fig 1f).

As mentioned above, each stimulus group from a given session of the passive data consists of 10 members, corresponding to the response vectors of a given 1-second block from 10 movie repeats (Fig 2a). Across stimulus groups, sessions, and mice (n_mice = 73), we find D to be small relative to the size of the neural state space D/n = 0.05 ± 0.04, but the variational space captures 91 ± 4% of the group’s variation (mean ± s.e.) [35, 36]. Note that the fact that the variational space is relatively small and yet captures a large variation in the data is not too surprising given m = 10, i.e. there are only 10 members of each stimulus group, and hence D could be at most 9. We find D = 2.7 ± 1.1 (mean ± s.e.). Below we will show these numbers are consistent with larger stimulus groups. We also find the mean along a given PC direction is strongly correlated with its percentage of variance explained (S1 Fig).

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Fig 2. Passive data: Feature space, drift, and drift’s dependence on time.

[a-d] Data from an exemplar mouse. (a) Response vectors (dots), mean response vectors (X’s), and first two PC dimensions (lines, scaled by variance explained), for two stimulus groups in a given session. Plotted in stimulus group 1’s PC space. (b) Same as previous subplot, but response vectors of stimulus group 1 across two different sessions, plotted in session 1’s PC space. (c) Pairwise angle between the response vectors of the 30 1-second time-blocks across the first five movie repeats of a single session. (d) Pairwise angle between mean response vectors across the three different sessions, same color scale as (c) (Methods). [e-h] Various metrics as a function of the time between earlier and later session, Δt, for all mice. All metrics are computed for each individual stimulus group, then averaged across all 30 groups. Colored curves are linear regression fits and shaded regions represent all fits within 95% confidence intervals of slope and intercept. (e) Average angle between mean response vectors. (f) Average (L2) magnitude of drift relative to magnitude of earlier session’s mean response vector. (g) Average change in variational space dimension, D, from later session to earlier session. (h) Average drift magnitude within earlier session’s variational space, ratio relative to full drift magnitude, see Eq (9). In yellow, the same metric if drift were randomly oriented in neural state space (mean ± s.e., across mice).

https://doi.org/10.1371/journal.pcbi.1010716.g002

Representational drift occurs between sessions.

We now consider how the stimulus groups representing the 1-second blocks of Natural Movie One drift from one session to another (Fig 2b). Since we have three distinct sessions for each mouse, we can analyze three separate instances of drift, 1 → 2, 2 → 3, and 1 → 3.

We first verify the difference in neuronal representations between sessions is distinct from the within-session variation. To do so, we train a linear support vector classifier (SVC) to distinguish response vectors of a given stimulus group from one session to another session. We compare the 5-fold cross validated accuracy of this SVC to two SVCs trained to distinguish members of the same stimulus group within a single session. This is done by creating two types of within-session subgroups: (1) subgroups correspond to the first or second half of the session (2) subgroups correspond to either the even or odd movie repeats. The SVC trained to distinguish separate sessions achieves an accuracy significantly higher than chance (68 ± 8%, mean ± s.e.), while both within-session accuracies are at chance levels (51 ± 7 for first-second half, 45 ± 5% for even-odd, mean ± s.e.). We also note that previous work found the mean activity rates, number of active cells, pupil area, running speed, and gradual deterioration of neuronal activity/tuning do not explain the variation between sessions [11].

Now let us quantify how the response vectors change as a function of time. Throughout this work, we use the angle between response vectors as a measure of their similarity. Across repeats but within the same session, we find the angle between response vectors corresponding to the same time-block to generally be small, i.e. more similar, relative to those belonging to different blocks (Fig 2c). Comparing the mean response vectors of stimulus groups across sessions, we find a smaller angle between the same group relative to different groups, but it is evident that the neuronal representation of some groups is changing across sessions, as shown by the greater angle between their mean response vectors (Fig 2d).

Drift has a weak dependence on the exact time difference between sessions.

As a measure of the size and direction of drift, we define the drift vector, d, as the difference in the mean response vector of a given stimulus group from one session to another (Fig 1f, Methods). Additionally, we denote the time difference between pairs of imaging sessions by Δt. We will always take Δt > 0, so we refer to the individual sessions between which we are measuring drift as the earlier and later sessions.

Recall that the number of days between sessions is mouse-dependent. In order to compare aggregate data across mice, we would like to better understand how certain features of drift change as a function of Δt. To this end, we compare how several characteristics of the stimulus groups change as a function of time between sessions (Methods). We see a very modest increase in the average angle between mean response vectors as a function of Δt (Fig 2e). This indicates that mean response vectors are, on average, only becoming slightly more dissimilar as a function of the time between sessions (< 1 degree/day). Many other geometric characteristics of the drift do not change considerably as a function of Δt as well. We note here that this result does not mean that drift at shorter timescales does not accumulate over time, nor does it mean longer timescale drift would also exhibit such a weak dependence on Δt. See the Discussion for further consideration of how this result compares to other works.

We see the magnitude of the drift vector, d, is on average slightly larger than that of the mean response vector (Fig 2f). This not only indicates that size of drift on the time scale of days is quite large, but also that the size of drift does not seem to be increasing considerably with the time difference between sessions. Across Δt, we also see very little change in the variational space dimension, D, indicating the size of the variational space is not changing considerably (Fig 2g). As a measure of the direction of the drift relative to a stimulus group’s variation space, we consider the ratio of the drift’s magnitude that lies within the earlier session’s variational space (Methods). Across Δt values, we find this is quite steadily around 0.5, meaning about half the drift vector’s magnitude lies within the relatively small variational space (Fig 2h). This is significantly higher than if the drift vector were simply randomly oriented in neural state space (Fig 2h).

We take the above metrics to indicate that the drift characteristics of excitatory cells in V1 are fairly steady across the Δt we investigate in this work, 1 to 9 days. Thus, in the remainder of this section we aggregate the data by session and across all Δt (see S1 Fig for additional plots as a function of Δt).

Drift’s dependence on variational space.

Seeing that the drift direction lies primarily in the earlier session’s variational space, next we aim to understand specifically what about said variational space determines the direction of drift. Since, by definition, the variational space is spanned the stimulus group’s PC vectors, this provides a natural basis to understand the properties of drift. Looking at the magnitude of drift along the earlier session’s PC direction as a function of the PC direction’s variance explained ratio, v_i, we find that the magnitude of drift increases with the PC direction’s variance explained (Fig 3a). That is, stimulus group directions that vary a lot tend to have the largest amount of drift. Additionally, there is a strong trend toward drift occurring at an angle obtuse to the top PC directions, i.e. those with high variance explained (Fig 3b). Said another way, drift has an increasingly large tendency to move in a direction opposite to PC directions with a large amount of variance Furthermore, we find both the magnitude and angular dependence of drift as a function of variance explained to be well fit by linear curves (Fig 3a and 3b).

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Fig 3. Passive data: Drift geometry and classifier persistence.

[a-c] How various drift metrics depend on PC dimension of the earlier session’s variational space. Metrics are plotted as a function of PC_i’s ratio of variance explained, v_i, across all stimulus groups. Colored curves are linear regression fits and shaded regions (often too small to see) are all fits within the 95% confidence intervals of slope and intercept. (a) Magnitude of drift along PC_i direction relative to full (L2) magnitude of drift. (b) Angle of drift with respect to PC_i direction. (c) Post-drift variance explained along PC_i direction, black dotted line is equality. Linear regression fit to log(var. exp.). [d-f] Various metrics and how they change between sessions. The darker dots/lines always show mean value with error bars of ± s.e. The lighter color dots show data from individual mice. (d) The variational space overlap between earlier and later stimulus groups, 0 ≤ Γ ≤ 1. The “–” marker indicates the average value of Γ for randomly oriented variational spaces of the same dimensions. (e) Angle between linear support vector classifiers (normal vector) trained on distinct sessions. The purple dotted line is the average angle between different sessions. (f) Cross classification accuracy as a function of trained data session (Class.) and tested data session (Data). The “–” marker shows average classification accuracy when SVCs are randomly rotated by same angle that separates respective sessions’ classifiers. (g) The relative cross accuracy, see Eq (14), as a function of the angle of a random SVC rotation. The purple dotted line is again the average angle found between drift sessions, also shown in (e).

https://doi.org/10.1371/journal.pcbi.1010716.g003

What is the net effect of the above geometric characteristics? On average, a drift opposite the direction of the top PC directions results in a reduction of the mean value along said directions. Since a PC direction’s magnitude and variation are correlated (S1 Fig), this also causes a reduction of the variation along said direction. This can be seen directly by plotting the variance explained along the earlier session’s PC directions before and after the drift (Fig 3c). We see a decrease in variance explained in the top PC directions (below the diagonal), and an increase in variance explained for lower PC directions (above the diagonal). So at the same time variance is flowing out of the top PC directions, we find additional directions of variation grow, often directions that had smaller variation to begin with, compensating for the loss of mean/variance. Thus the net effect of this drift is to reduce the stimulus group variance along directions that already vary significantly within the group and grow variation along new directions.

A byproduct of this behavior is that the variational space of a stimulus group should change as it drifts. To quantitatively measure the change in variational spaces, we define the variational space overlap, Γ (see Methods for precise definition). By definition, 0 ≤ Γ ≤ 1, where Γ = 0 when the variational spaces of the earlier and later sessions are orthogonal and Γ = 1 when the variational spaces are the same (and when one space is a subspace of the other, see Methods). Between all sessions, we find Γ ≈ 0.5, which is not far from Γ values if the subspaces were simply randomly oriented, indicating that the variational space of a given stimulus group indeed changes quite a bit as it drifts (Fig 3d).

Classifier persistence under drift.

Above we showed that drift is both large and has a preference toward turning over directions with large variation, thus significantly changing a stimulus group’s variational space. Intuitively, this is difficult to reconcile with previous results (and results later in this paper) that have observed mice performance remains persistent despite a large amount of drift [6, 8, 9]. To quantify the separability of stimulus groups and how this changes under drift, for each session we train a linear SVC to distinguish groups within said session using 10-fold cross validation. In particular, to avoid feature space similarity due to temporal correlation in the movie, we train our SVCs to distinguish the response vectors from the first and last 1-second blocks of a given session.

For a given mouse, we find the linear SVCs trained on its sessions are significantly different from one another, as measured by the angle between their normal vectors, on average 57 degrees (Fig 2e). However, despite this difference, we find that when we use one session’s SVC to classify response data from a different session, the accuracy on the data does not fall significantly (Fig 2f). Interestingly, this appears to be a result of the high-dimensionality of the neural state space, as has been discussed previously [8]. Randomly rotating our SVCs in neural state space by the same angles we found between SVCs of different sessions, we achieve only slightly lower accuaracies (Fig 2f). Indeed, calculating the ratio of accuaracies as a function of the angle of the random rotation, we observe a monotonically decreasing function that is relatively stable up to the average angle we observed experimentally (Fig 2g). Note the relative cross accuracy is still finite at a change in the SVC angle of 180° because the SVC does not achieve 100% classification accuracy, so even with the weights flipped the SVC gets a non-trivial number of examples correct. We find it interesting that drift occurs such that the angle between SVCs is large yet not large enough to cause a significant dip in accuracy when used across sessions. Although they investigate drift in the parietial cortex, Ref. [8] also finds population decoding accuracies to be relatively stable across days for the part of the task that yields the high decoding accuracy. However, neural recordings at other parts of the tasks and smaller cell counts appear less stable. In the olfactory cortex, Ref. [9] finds a significant degradation in classification accuracy over time, though their decoders are tasked with a much more difficult 8-way classification and smaller cell counts.

Experimental setup and neuronal response.

Mice are continually presented one of eight natural images and are trained to detect when the presented image changes by responding with a lick (Fig 4a). After reaching a certain performance threshold on a set of eight training images, their neuronal responses while performing the task are measured over several sessions using in vivo 2-photon calcium imaging. Specifically, their neuronal responses are first imaged over two “familiar” sessions, F1 and F2, in which they must detect changes on the set of eight training images. Afterwards, two additional “novel” imaging sessions, N1 and N2, are recorded where the mice are exposed to a new set of eight images but otherwise the task remains the same (Fig 4b, Methods). Similar to the passive data, the time difference between pairs of imaging sessions, F1-F2 or N1-N2, is on the order of several days, but differs for each mouse.

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Fig 4. Behavioral data: Experimental setup and drift geometry.

(a) Summary of experimental setup. (b) Summary of session ordering, trial types, and extraction of response vectors from dF/F values. Bottom plot shows dF/F values over time, with colored columns representing image flashes where different colors are different images. [c-e] Various drift metrics of Hit trials and their dependence on PC_i direction of the earlier session’s variational space. Dark colors correspond to drift between familiar sessions, while lighter colors are those between novel sessions. Metrics are plotted as a function of each PC_i’s ratio of variance explained, v_i. Colored curves are again linear regression fits. (c) Magnitude of drift along a given PC_i direction, relative to full magnitude of drift. (d) Angle of drift with respect to PC_i direction. (e) Post-drift variance explained along PC_i direction (dotted line is equality). Linear regression fit to log(var. exp). [f-h] Various metrics as a function of session(s). Dark solid dots/lines show mean values with ± s.e. Light colored dots/lines show raw mice data. (f) Mean performance metric over engaged trails, d′ (Methods). (g) Angle between SVC normal vectors. (h) Cross classification accuracy, as a function of trained data session (Class.) and tested data session (Data). The “–” marker again shows average classification accuracy when SVCs are randomly rotated by same angle that separates respective sessions’ classifiers.

https://doi.org/10.1371/journal.pcbi.1010716.g004

We will be particularly interested in the neuronal responses of a mouse’s success and failures to identify an image change, which we refer to as “Hit” and “Miss” trials, respectively (see Methods for a parallel study of “Change” and “No Change” stimulus groups, which yields qualitatively similar results to what we present here). We once again form a response vector of the neuronal responses by averaging dF/F values across time windows for each neuron. The time window is chosen to be the 600 ms experimentally-defined “response window” after an image change occurs (Fig 4b, Methods). Once again, we define the response vector to only contain cells that are identified in both of the sessions that we wish to compare. Furthermore, to ensure a mouse’s engagement with the task, we only analyze trials in which a mouse’s running success rate is above a given threshold (Methods).

Since each session contains hundreds of image change trials, we have many members of the Hit and Miss stimulus groups. We once again find the dimension of the feature space representations, D, to be small relative to the size of the neural state space. Specifically, D/n = 0.06 ± 0.04 and 0.06 ± 0.05, yet it captures a significant amount of variation in the data, 0.79 ± 0.06 and 0.80 ± 0.06, for the Hit group of the familiar (n_mice = 28) and novel (n_mice = 23) sessions, respectively (mean ± s.e.) [35, 36]. Additionally, we continue to observe a strong correlation between the mean along a given PC direction and its corresponding variance explained (S2 Fig).

Drift geometry is qualitatively the same as the passive data.

Once again, we distinguish the the between-session variation due to drift from the inter-session variation by training linear SVCs. We again find the SVCs trained to distinguish sessions achieve an accuracy significantly higher than chance. For example, in the familiar session drift, the Hit stimulus groups can be distinguished with accuracy 74 ± 11% (mean ± s.e., novel data yields similar values). Meanwhile, the SVCs trained to distinguish the first/second half of trials and the even/odd trials within a single session do not do statistically better than chance (familiar Hit groups, first/second: 57 ± 8%, even/odd: 52 ± 6%, mean ± s.e.). Additional quality control checks were performed on this dataset to ensure behavioral differences and imaging variability across days could not explain the drift we observed (Secs. 4.3.1 and 4.3.2, S7, S8, and S9 Figs).

Similar to the passive data, the exact number of days between F1 and F2, as well as N1 and N2, differs for each mouse. Across Δt values, we find many of the same drift characteristics we observed in the passive data, including: (1) a drift magnitude of the same order as the magnitude of the mean response vector, (2) on average, no change in the size of the variational space, and (3) a greater than chance percentage of the drift vector’s magnitude lying within the earlier session’s variation space (S2 Fig). Across all these measures, we do not find a significant quantitative dependence on Δt, for the range of 1 to 6 days, so we will continue to treat drift data from different mice on equal footing, as we did for the passive data.

Between both the familiar and novel sessions, we again find the magnitude of drift along a given PC direction is strongly correlated with the amount of stimulus group variation in said direction (Fig 4c). Although their ratio is comparable to the familiar sessions, both the magnitude of drift and the mean response vectors are significantly larger in the novel sessions, consistent with previous findings (S2 Fig). Additionally, for both pairs of sessions, the drift again has a tendency to be directed away from the PC directions of largest variation (Fig 4d). The net effect of these characteristics is that we once again observe a flow of variation out of the top PC directions into directions that previously contained little variation (Fig 4e). It is fascinating that the familiarity/novelty of the image set does not seem to significantly affect the quantitative characteristics of these three measures.

Inter-group characteristics under drift.

Does the drift affect the mouse’s performance? We observe no statistically significant difference in the performance of the mice despite the large amount of drift between sessions (Fig 4f). Notably, the novelty of the second image set does not affect the performance of the mice either, showing their ability to immediately generalize to the new examples.

We train a linear SVC to distinguish the Hit and Miss stimulus groups within a given session using 10-fold cross validation. Comparing the SVC between pairs of familiar/novel sessions, we again observe a significant amount of change between the SVCs as measured by the angle between their normal vectors (Fig 4g). Once again, an SVC trained on the earlier (later) session is able to classify the later (earlier) session’s data with accuracy comparable to the classifier train on the data itself (Fig 4h). These are the same results we saw on the passive data: despite significant changes in the SVC, the stimulus groups do not seem to drift in such a way as to significantly change their linear separability.

One hypothesis for the persistence of performance under drift is that individual stimulus groups drift in a coordinated manner [11, 26, 29] (though some studies see a significant lack of coordination [9]). We find the average angle between the drift vectors of the Hit and Miss groups to be 68.5 ± 16.5° and 56.9 ± 14.6° for familiar and novel sessions, respectively (mean ± s.e.). That is, the drift directions are aligned at a level greater than chance (on average 90° for two random vectors), indicting that there is at least some level of coordination between the individual stimulus group drifts. Since we have found a tendency for drift to lie in the earlier session’s variational space, an alignment in drift could be a byproduct of a similarity of the two groups’ variational spaces. Indeed, we find the variational subspaces of the two stimulus groups to be aligned with one another at a rate significantly higher than chance, as measured by the variational space overlap, Γ (S2 Fig).

Experimental setup.

We train our CNNs on the CIFAR-10 dataset consisting of 60, 000 32 × 32 color images from 10 different classes (e.g. birds, frogs, cars, etc.) [38]. Once a steady accuracy is achieved, we analyze the the time-evolution of the feature space representations under continued training of a two-class subset (results are not strongly dependent upon the particular subset). We take the analog of the response vectors of the previous sections to be the n-dimensional feature vector in the feature (penultimate) layer and the separate stimulus groups to be the classes of CIFAR-10. Throughout this section, we train all networks with stochastic gradient descent (SGD) at a constant learning rate and L2 regularization (Methods). Finally, we note that our goal in training the CNNs is to arrive at a feature space that somewhat resembles the neural data in the sense that the distinct groups, i.e. the CIFAR-10 classes, occupy somewhat distinct subspaces and the network’s performance had stabilized. Specifically, we confirm the networks achieve stable accuracies well above chance, but do not optimize the networks’ hyperparameters or learning schedules to achieve the highest possible accuracy for the given architecture.

Different types of noise to induce drift.

We begin by training our CNNs in a setting with minimal noise: the only element of stochasticity in the network’s training is that due to batch sampling in SGD. Once our networks reach a steady accuracy, under continued training we observe very little drift in the feature space representations (red curve, Fig 5b). To induce feature space drift, we apply one of five types of additional noise to the feature layer of our networks:

1. Additive node: Randomness injected directly into the feature space by adding iid Gaussian noise, , to each preactivation of the feature layer.
2. Additive gradient: Noise injected into the weight updates by adding iid Gaussian noise, , to the gradients of the feature layer. Specifically, we add noise only to the gradients of the weights feeding into the feature layer. This is similar to how noise was injected into the Hebbian/anti-Hebbian networks studied in Ref. [29].
3. Node dropout: Each node in the feature layer is omitted from the network with probability p [19, 39].
4. Weight dropout: Each weight feeding into the feature nodes is omitted from the network with probability p [40].
5. Multiplicative node: Each feature node value is multiplied by iid Gaussian noise, . This is also known as multiplicative Gaussian noise [39]. Note this type of noise is often seen as a generalization of node dropout, since instead of multiplying each node by ∼Bernoulli(p), it multiplies by Gaussian noise.

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Fig 5. Artificial neural networks: Hyperparameter fits and drift geometry as a function of Δt and variance explained.

(a) Measure of fit to experimental data, Z_total see Eq (19), as a function of noise hyperparameters, p (top labels) or σ (bottom labels). Dots are best fits, for which additional data is plotted here and in supplemental figures (S5 Fig, Methods). [b-c] Various metrics as a function of the time between earlier and later session, Δt. Colored curves are linear regression fits. All data is averaged over 10 initializations. (b) Average magnitude of drift relative to magnitude of mean response vector. (c) Average percent of drift vector that lies in the variational space of initial session. [d-i] Various drift metrics and their dependence on PC dimension of the earlier session’s variational space. Metrics are plotted as a function each PC_i’s ratio of variance explained, v_i, of the corresponding stimulus group Colored curves are linear regression fits. Grey curves are behavioral data fits from the novel sessions shown in Fig 4c, 4d and 4e. Middle row is for networks with additive Gaussian noise (σ = 0.1) and bottom row is with node dropout (p = 0.5). All data is averaged over 10 initializations. (d, g) Magnitude of drift along PC_i direction, relative to full magnitude of drift. (e, h) Angle of drift with respect to PC_i direction. (f, i) Post-drift variance explained along PC_i direction (dotted line is equality). Linear regression fit to log(var. exp.).

https://doi.org/10.1371/journal.pcbi.1010716.g005

All types of noise are applied both during initial training and the drift observation at steady accuracy. Of these five types of noise injection, below we will show that node dropout, and to a lesser extent multiplicative node and weight dropout, induce drift that strongly resembles that which we observed in both the passive and behavioral data.

Changes in the feature space are of course dependent upon the time difference over which said changes are measured. Similar to the experimental results above, below we will show that many of the drift metrics discussed in this section show no significant dependence upon Δt, so long as Δt is large compared to the time scale of the noise. Additionally, the degree of drift found in the feature space of our CNNs is dependent upon the size of the noise injected, i.e. the exact values of σ and p above. For each type of noise, we conducted a hyperparameter search over values of p (0.1 to 0.9) or σ (10⁻³ to 10⁺¹) to find the values that best fit the experimental data (Fig 5a, Methods). Below we discuss results for the best fits of each types of noise. We find the qualitative results are not strongly dependent upon the exact values chosen, up to when too much noise is injected and the network does not train well.

Feature space geometry and Δt dependence.

Once more, we find the variational space of the various classes to be small relative to the neural state space. For example, under p = 0.5 node dropout we find D/n = 0.070 ± 0.003, capturing 82.2 ± 0.3% of the variance. Notably, the feature space geometry continues to exhibit a correlation between variance explained and the mean value along a given direction, again indicating that directions which vary a lot tend to have larger mean values (S4 Fig). Additionally, the variational spaces of the different classes continue to be aligned at a rate greater than chance (S4 Fig).

As expected, all five noise types are capable of inducing drift in the representations. This drift occurs amongst stable accuracies, that are comparable across all types of noise and steady as a function of time (S4 Fig). We find the size of drift relative to the size of the means to be comparable to that which was found in experiments for several types of noise (Fig 5b). Additionally, the relative magnitude of the drift for all types of noise is close to constant as a function of Δt. Similar to the experimental data, we find all the drifts do not induce a significant change in the dimensionality of the variational space (S4 Fig). Finally, we again note that the drift percentage that lies in variational space for all types of noise is significantly larger than chance, though all but the smallest drifts have a ratio smaller than that observed in experiment (Fig 5c).

Having observed several metrics that are constant in Δt, we use Δt = 1/10 epoch henceforth since it is within this weak Δt-dependence regime, and thus comparable to the experimental data analyzed in the previous sections. Decreasing the time scale of the noise injection to slower than each forward pass, it is possible to observe stronger Δt dependence (Fig 6f). However, since our goal in this work is to match onto experimental data, where there we find evidence for very weak Δt dependence, here we will focus on results within said regime.

Dropout drift geometry resembles experimental data.

For clarity, here in the main text we only plot data/fits for the additive node and node dropout noises (Fig 5). Equivalent plots for SGD only and all other types of noise, as well as a plot with all six fits together, can be found in the supplemental figures (S5 Fig).

For all types of noise, we find an increasing amount of drift with PC dimension/variance explained, though the overall magnitude and distribution over variance explained vary with the type of noise (Fig 5d and 5g). All types of noise also exhibits a degree of increasing angle between the drift direction as a function of variance explained. However, for several types of noise, this trend is very weak compared to the experimental data, and it is clear the fitted data is qualitatively different from that observed in experiment (Fig 5e). The exceptions to this are node dropout, and to a lesser degree, weight dropout, where the fits and raw data match experiment quite well (Fig 5h). One way this can be seen quantitatively is by comparing the r-values of the linear fits, for which we see the node dropout data is relatively well approximated by the linear trend we also saw in the experimental data (S5 Fig). We see all types of noise result in a flow of variance out of the top PC dimensions and into lower dimensions. Once again though, for many types of noise, the amount of flow out of the top PC dimensions is very weak relative to the passive and behavioral data (Fig 5f). We do however see that the two types of dropout, as well as multiplicative node, all exhibit a strong flow of variation out of directions with large variations (Fig 5i and S5 Fig). Finally, it can also be helpful to compare drift geometry as a function of the earlier session’s PC dimension instead of variance explained. See S6 Fig for plots of both experimental setups and all six ANN setups considered here.

From these results, we conclude that the three types of dropout, especially node dropout, exhibit features that are qualitatively similar to experiments. We now turn our focus to additional results under node dropout noise.

Classifier and readouts persistence under drift.

Unique to the ANNs in this work, we have access to true readouts from the feature space to understand how the representations gets translated to the network’s ultimate output. Previously, we fit SVCs to the feature space representation to understand how drift might affect performance, so to analyze our CNNs on an equal footing, we do the same here. Once again, we find classifiers fit at different time steps to be fairly misaligned from one another, on average 71 (±3) degrees (mean ± s.e., Fig 6a). Despite this, an SVC trained at one time step has slightly lower yet still comparable accuracy when used on feature space representations from another time step, with relative cross accuracy is 0.86 (±0.03) (Fig 6b). This is similar to what we observed in both the experimental datasets.

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Fig 6. Artificial neural networks: Additional properties of drift geometry.

[a-d] Various quantities as a function of relative/absolute training time (in epochs). Means are shown as dark lines, with 95% confidence intervals shaded behind. Raw data is scattered behind. (a) Angle between SVC classifiers (normal vectors) as a function of the time difference. The grey dashed line is the average for the novel Hit data shown in Fig 4g. (b) Cross classification accuracy as a function of time difference between classifier training time (Class.) and testing time (Data). (c) Difference in angle of a stimulus group’s readout as a function of the time difference. Note the different vertical scale from (a). (d) Deviation of the angle between a stimulus group’s drift and the respective readout from perpendicular (i.e. 90 degrees). The dashed green line is the average across time. The dotted black line is the angle between two randomly drawn vectors in a feature space of the same dimension. (e) Fits of variance explained versus angle of drift with respect to PC direction for regular node dropout (purple), targeted maximum variance node dropout (pink), and targeted minimum variance node dropout (yellow). The inset shows the r-values of the respective fits. (f) Difference in response vector angle as a function of Δt. The dashed vertical line indicates the time scale on which the node dropouts are updated (1/epoch).

https://doi.org/10.1371/journal.pcbi.1010716.g006

Interestingly, when we look at how the readouts of the CNN change with time, we see their direction changes very little, on average only 2.6 degrees over 5 epochs (Fig 6c). How can this be despite the large amount of drift present in the network? Comparing the direction of the drift to the network’s readouts, we see that they are very close to perpendicular across time (Fig 6d). If the stimulus group means move perpendicular to the readouts then, on average, the readout value of the group remains unchanged. As such, despite the stimulus group drift being large, on average it does not change the classification results. Perhaps contradicting that this is a result of special design, we find the average angle between the drift and readouts to be consistent with chance, i.e. if the drift direction were simply drawn at random in the high-dimensional feature space. Thus we cannot rule out that the ability for the readout to almost remain constant in the presence of large drift is simply a result of the low probability of drift occurring in a direction that significantly changes the network’s readout values in high dimensions [24]. Notably, we see comparatively more drift in the readouts for some other types of noise. For example, gradient noise causes a drift in the readouts of 18.6 degrees over 5 epochs.

Additional drift properties in ANNs.

Having established that the noise from node dropout strongly resembles that found in our earlier data, we now use this setting to gain some additional insights behind the potential mechanisms of representational drift.

Although we find our CNNs and experiments have comparable drift magnitude relative to the size of their mean response vectors, the CNNs appear to have significantly more variability due to drift compared to in-class variance than our experimental setups. For node dropout, we find SVCs trained to distinguish data from time steps separated by Δt = 1/10 epoch achieve perfect accuracy across trials, indicating the stimulus groups are linearly separable. SVCs trained to distinguish even/odd examples within a single class have chance accuracy, 49.3 ± 0.8% (mean ± s.e.), similar to experiment. The CNN also exhibits a coordination of drift between the two sub-groups of interest, whose drift vectors are separated by 40.4 ± 4.9 degrees (mean ± s.e.). As mentioned earlier, we also continue to observe a greater-than-chance variational space overlap between said stimulus groups (S4 Fig).

Next, we would like to see if we can further pinpoint what about node dropout causes the drift. To this end, we define a type of targeted node dropout algorithm that preferentially targets nodes with high variation (Methods). We find that qualitatively and quantitatively, targeted node dropout also has similar drift characteristics to the experimental data (Fig 6e and S4 Fig). Furthermore, this results holds for a smaller number of averaged nodes dropped per dropout pass, on average only 17 nodes per pass as compared to regular node dropout which drops np = 42 nodes per pass. Of course, with dropout percentages used in practice on the order of p = 0.5, the nodes that vary the most will be dropped quite frequently, so its not surprising that we are observing similar results here. If instead we target the nodes with the smallest variation, we do not observe a significant amount of drift or the characteristics we find in the experimental data, despite dropping the same number of nodes on average (yellow curve, Fig 6e and S4 Fig). Altogether, this suggests that it may be the dropping out of large variance/mean nodes that causes the characteristics of drift that we are observing.

In the above noise setups, we only injected additional noise in the feature layer, including the weights directly prior to said layer. We find that if dropout is also applied to earlier layers within the network, qualitatively similar drift characteristics continue to be observed (S4 Fig). However, when dropout was removed from the feature layer and applied only to an earlier layer, the amount of drift in the feature layer dropped significantly (S4 Fig).

In this work, we have focused on drift in the regime where certain metrics are close to constant as a function of Δt. As a means of verifying the transition into this regime, we can see if drift in ANNs is different on shorter time scales. To reach a regime where drift occurs slowly we lengthen the time scale of noise injection via node dropout by reducing the frequency of when the network recalculates which nodes are dropped, which is usually done every forward pass. When we do this, we observe a sharp transition in average angle between response vectors as a function of Δt when it is above/below the noise-injection time scale (Fig 6f). We leave further observations of the differences in drift properties at such timescales for future work.

Feature and variational space.

For stimulus group p in session s, we define the number of members of said stimulus group to be . Let the n-dimensional response vector be denoted by where indexes members of said stimulus group. The mean response vector for stimulus group p in session s is then (2) The dimension of PCA on stimulus group p in session s is . Denote the (unit-magnitude) PC_i vector by and the corresponding ratio of variance explained as for (alternatively, w_i and v_i for brevity). PCs are ordered in the usual way, such that v_i ≥ v_j for i < j. We remove the directional ambiguity of the PC directions by defining the mean response vector of the corresponding stimulus group and session to have positive components along all PC directions. That is (suppressing explicit s and p dependence), (3) If , then we simply redefine the PC vector to be w_i → −w_i. The dimension of the feature space representation is the “participation ratio” of the PCA variance explained ratios (4) To build some intuition for this quantity, note that only when for all , i.e. the variance is evenly distributed amongst all PC dimensions. Additionally, only when and for all i > 1, i.e. all the variance is in the first PC dimension. This measure is often used as a measure of subspace dimensionality [62–64]. Also note this quantity is invariant up to an overall rescaling of all v_i, so it would be unchanged if one used the variance explained in each PC dimension instead of the ratio of variance explained. In general, this quantity is not an integer. Finally, the variational space of stimulus group p in session s is defined to be the subspace spanned by the first PC vectors of said stimulus group and session, for , with ⌈⋅⌉ the ceiling function. As we showed in the main text, the dimension of the feature space, Eq (4), is often a small fraction of the neural state space, yet the variational space contains the majority of the variance explained.

We note that recently a similar quantification of feature spaces found success in analytically estimating error rates of few-shot learning [64].

Drift between sessions.

We define the drift vector of stimulus group p from session s to session s′ to be (5) Note that the ordering of indices in the superscript matter, they are (earlier session, later session), and will be important for later definitions. For this vector to be well-defined, and must be the same dimension. This is always the case in the main text since we restrict to the subset of neurons that are identified in the sessions that we wish to compare. If the time of sessions s and s′ are respectively t^s and , we define the time difference between sessions s and s′ by , dropping the superscripts in the main text. Note that if a neuron is not identified in a session it could either be because (1) the cell has left the imaging field or (2) the cell is inactive during the entire session and thus cannot be detected via calcium imaging. Although we wish to control for the former, the two cases were not distinguished in the datasets analyzed in this work and thus this methodology misses neurons that are completely inactive one session and active in another session.

We use the above geometric quantification for the passive data, active data, and artificial neural networks.

1. For the passive data, p = 1, …, 30 corresponds to the non-overlapping one-second time-blocks of the movie. Meanwhile, for all p and s since each session has ten movie repeats. Finally, s = 1, 2, 3 corresponds to the three different sessions over which each mouse is imaged.
2. In the behavioral data, we have p = Hit, Miss and s = F1, F2, N1, N2. In the supplemental figures, we also consider p = Change, No Change, see Sec. 4.3 below for details (S3 Fig). The number of examples for each stimulus group is the number of engaged trials in a given session, so differs over both stimulus groups and sessions.
3. For the artificial neural networks, p = cat, dog, car, …, the distinct classes of CIFAR-10. In this case, s represents the different time steps during training when the feature space measurements are taken. In this work we only consider s values for which the the accuracy of the network is steady (see below for additional details). In practice, we use test sets where for all stimulus groups p.

Geometric measures.

In this work, we often use the angle between two response vectors as a measure of similarity. This is different than the results of Ref. [11], where Pearson’s correlation coefficient (see below) is used as a similarity measure between response vectors. Here we use angle because (1) our goal is to explore geometrical characteristics of drift, and we believe angle is more interpretable in this context than Pearson’s correlation and (2) angle is a rotationally invariant metric, and thus allows for comparisons independent of a mouse’s particular basis of neural state space. Note neither of these measures are sensitive to the magnitude of the response vectors being compared. The two measures yield qualitatively similar results (S1 Fig).

The angle (in degrees) between two n-dimensional vectors x and y is defined as usual, (6) where ‖⋅‖₂ is the L2-normalization and 0 ≤ θ ≤ 180. Although the Pearson correlation coefficient is not used in this work, it is used in other works [11] as a quantitative measure of representation vector similarity. For the purpose of comparison, we reproduce the expression here, (7) where −1 ≤ r ≤ 1 and and . From this expression we see the Pearson correlation coefficient is fairly similar to the angular measure, up the arccos(⋅) mapping and the centering of vector components. Note neither θ nor r are dependent upon the individual vector magnitudes, but the former is invariant under rotations of neural state space and the latter is not.

Often it will be useful to have a metric to quantify the relation between a drift vector to an entire subspace, namely a variational subspace. To this end, let the projection matrix into the variational subspace of stimulus group p and session s be . can be constructed from the PCs of stimulus group p in session s. Let be the matrix constructed from the first (orthonormal) PCs of stimulus group p in session s, (8) The column space of is the variational space of stimulus group p in session s. Then the projection matrix is . The ratio of the drift vector that lies in the variational subspace of stimulus group p in session s, or the drift in variation ratio, is defined to be (9a) (9b) where and the second line follows from the fact the form an orthonormal basis of the variational subspace. Intuitively, this quantity tells us how much of the drift vector lies in the variational space of stimulus group p. This is done by projecting the vector d into the subspace, and comparing the squared L2 magnitude of the projected vector to the original vector. If the drift vector lies entirely within the subspace, . Meanwhile, if the drift vector is orthogonal to the subspace, .

Finally, it will be useful to compare two variational subspaces to one another. To this end, we define the variational space overlap of stimulus group p between sessions s and s′ to be (10a) (10b) where and in the second line we have used the fact the are an orthonormal basis. Similar measures of subspace similarity are explored in Refs. [65, 66]. There it is also argued such measures are rotationally invariant to the orthonormal basis spanning either of the subspaces.

From the first line, we see is simply a sum of the drift in variation ratio, Eq (9), of each basis vector of the variational space of session s′ relative to the variational space of s, weighted by the size of the smaller of the two variational spaces. From the second line, we see this measure is equivalent to a sum of the pairwise squared dot products between the PCs of the two variational subspaces, again weighted by the dimension of the smaller subspace. Additionally, from the second line it is clear that . It is straightforward to show if one variational space is a subspace of (or equal to) the other variational space. Additionally, if the two subspaces are orthogonal. As another example, if both variational subspaces were of dimension 6, and they shared an subspace but were otherwise orthogonal, then . In the S1 Appendix, we show this metric is also invariant under changes in the orthonormal basis spanning the subspaces. A quick way to argue this invariance is to notice in the definition of , Eq (10a), the magnitude of the projection of to the subspace s is invariant under the rotation of the orthonormal basis of s. Additionally, from Eq (10b), is symmetric with respect to the subspaces s and s′, so if it is invariant under rotations of s this must also be true for s′.

Fig 2 details.

Fig 2a was generated by projecting the response vectors for two stimulus groups onto the first two PC components of the first group. Similarly, Fig 2b was generated by projecting the response vectors for a given stimulus group from two different sessions onto the first two PC components of the group in the first session. Fig 2c simply consists of the pairwise angle between the response vectors of the first five movie repeats for an exemplar mouse. Since each repeat consists of 30 response vectors, in total there are (5 × 30)² points shown. Fig 2d is generated by first finding the mean response vector, , for each stimulus group p = 1, …, 30. We then compute the pairwise angle within the session and between sessions, for s, s′ = 1, 2, 3.

Fig 2e was generated using the same mean response vectors for each stimulus group, where now the pairwise angle between the mean response vectors belonging to the same group are computed. This is done for all 30 stimulus group and then the average of all between-session angles is computed, i.e. (11) This quantity is plotted with its corresponding Δt and then fitted using linear regression. Error bars for all linear regression plots throughout this work are generated by computing the 95% confidence intervals for the slope and intercept separately, and then filling in all linear fits that lie between the extreme ends of said intervals. Fig 2f, 2g and 2h are all computed similarly, the drift metric for a given division is computed and then, for each mouse and separate instance of drift, the quantity is averaged over all 30 stimulus groups. In particular, the quantities plotted in Fig 2f, 2g and 2h are respectively (12a) (12b) (12c) as a function of Δt between the earlier session s and later session s′, see Sec. 4.1 above for definitions of the quantities on the right-hand side of these expressions. Note the distinction of s and s′ matter for these quantities. Finally, for Fig 2h the chance percentage for drift randomly oriented in neural state space was computed analytically. For a randomly oriented drift vector projected onto a dimensional subspace, . This quantity was averaged over stimulus groups and the three distinct drifts, before the average and standard error was computed across mice.

Fig 3 details.

The scatter plots in Fig 3a, 3b and 3c are averaged over p and plotted for each mouse and all three session-to-session drifts, i.e. 1 → 2, 2 → 3, and 1 → 3, as a function of the corresponding v_i. Any plot data with v_i ≤ 10⁻⁵ was omitted. For Fig 3a, 3b and 3c, they are respectively (13a) (13b) (13c) where is the percent of variance explained of stimulus group p in session s′ along the ith PC dimension of stimulus group p in session s (i.e., ). We found linear fits on the variance explained versus drift magnitude and PC-drift angle were better using a linear variance explained scale versus logarithmic variance explained scale.

For Fig 3d, we computed the , Eq (1), between the pair of variational spaces of the three possible pairs of sessions. This was done for each stimulus group p and mouse, then the raw mouse data for each pair of sessions is the average value of across all p. The average and ± s.e. bars were then computed across all mice. To generate the data in Fig 3e and 3f, we trained a linear SVC to distinguish the stimulus groups belonging to the first and last 1-second time-blocks of the movie. The first and last 1-second time-blocks were chosen to avoid any temporal correlation in the similarity of the movie scenes. For each mouse and each session, we trained our SVCs using 10-fold cross validation with an L2 penalty. The SVCs achieve an accuracy well above chance, 79 ± 15% (mean ± s.e.). For all folds belonging to sessions s and s′, pairwise angles between their normal vectors were determined using Eq (6) before averaging over all 10 folds.

Finally, we wanted to investigate if there was something special about the orientation of the SVCs between sessions that allowed them to have large angular separation by still high cross-classification accuracy. To this end, we tested their accuracy relative to random rotations of SVCs. For a given pair of sessions s and s′, we found the angle between the normal vectors of their respective SVCs. Using said angle, we generated a random rotation direction in neural state space and rotated the weights of the SVC of session s′ by the same angle. Let the accuracy of the SVC trained on data from session s, classifying the data from session s′, be denoted by . We define the relative cross correlation accuracy to be the accuracy of the cross-classification relative to the accuracy of classifier trained on the data, i.e. (14) By definition, . We computed the relative cross correlation accuracy (again using 10-fold cross validation) of the randomly rotated SVC relative to that of session s′ and did this for 10 separate random rotations (where the angle is fixed, but the rotation direction varies). In practice, we found these randomly rotated SVCs only had slightly lower accuracy than the SVCs actually trained on the separate sessions (shown as “–” marks in Fig 3f). The plot shown in Fig 3g was generated in the exact same manner of randomly rotating SVCs, but now the angle is a set value instead of it being equal to the angle between respective sessions’ SVCs. Note the cross-classification accuracy does not drop to 0 when the SVC is rotated 180 degrees (i.e. completely flipped) because, in general, the SVCs do not achieve 100% classification accuracy, so even the flipped SVC gets some examples correct (additionally, we are not changing the bias term).

Fig 4 details.

Many of the plots in Fig 4 were generated analogously to those in Fig 3, so refer to Sec. 4.2 above for details. A notable exception is that, due to the comparatively large stimulus group sizes, drift data from the two groups was not averaged over as it was across the 30 stimulus groups in the passive data. The performance metrics shown in Fig 4c, 4d and 4e are only for the Hit stimulus group, see S3 Fig for other groups. Once again, linear fits on the variance explained versus drift magnitude and PC-drift angle were better using a linear variance explained scale versus logarithmic variance explained scale. The performance metrics shown in Fig 4f make use of an experimentally-defined performance metric, d′, averaged across all engaged trials, see Ref. [33] for additional details. Specifically, (15) where Φ⁻¹(⋅) is the inverse cumulative Gaussian distribution function and R_H and R_F are the running Hit and False Alarm rates, respectively. Finally, Fig 4g and 4h were generated analogously to Fig 3e and 3f, respectively.

S2 Fig details.

For S2(b) Fig, variational space overlaps were computed between two stimulus groups using Eq (10). Values for randomly oriented variational spaces were analytically approximated to be . To arrive at this expression, without loss of generality take the smaller of the two subspace to span the first D directions of neural state space. We can construct the larger subspace, with dimension D′, by randomly drawing D′ unit vectors in neural state space sequentially, ensuring they remain orthogonal. The first of these has an average γ of D/n with respect to the smaller neuronal subspace. Since the second basis vector must be drawn from the subspace orthogonal to the first, it has an average γ of D/(n − 1). This process repeats to the last basis vector, with γ = D/(n − D′). In practice, since n ≫ D′, we neglect the numerical difference and simply approximate γ for all D′ basis vectors to simply be D/n. Then, summing together each γ and dividing by the D, we arrive at our aforementioned expression.

4.3.1 Behavior control.

It is well known that behavior can influence neuronal responses and their stability [15, 26, 68]. In this section, we discuss additional checks that were performed on the behavioral data to ensure that the properties of representational drift that we observe cannot be explained by the behavior of the mice. The influence of behavior on the passive data has been analyzed previously in Ref. [11], where they found behavioral metrics such as running speed and pupil area could not explain the drift they observed. Additionally, recent work has highlighted behavioral contributions to within-session drift, i.e. drift on the order of hours [25]. Notably, here we are concerned with drift on longer time scales (days) and purposely analyzes data that is averaged over entire sessions in an attempt to remove any latent effects such as behavioral dependence.

We will be specifically concerned with the behavioral metrics of running speed, pupil area, and eye position throughout this section. All three of these metrics were available for all but one mouse that was used only in the familiar analysis.

Since we are only concerned with the mice’s representations during the response window of Hit and Miss trials during which they are engaged with the task (see above), we similarly filter the behavioral metrics to these subsets of time points. Additionally, behavioral metrics are averaged over the response windows in the same way as the response vectors. Altogether, this yields a single quantity for the three behavioral metrics for each Hit and Miss trial (S7(a) Fig). Filtering to the subset of engaged response windows and also averaging within the response window yields a more narrow distribution of the behavior metrics compared to the raw data across the entire session (S7(b) Fig).

To begin, we investigated how much the behavioral metrics could explain the size of the response vector. For each mouse, we fitted the magnitude of the response vector of a given trial using a 4d linear regression (the three behavioral metrics, with eye position being two numbers) and indeed found significant evidence that said metrics influence the response vectors for some of the mice (F-test, S7(c) and S7(f) Fig). As mentioned above, the fact that behavior can influence neuronal responses is well known [15, 26, 68], but what is important for our purposes is that differences in behavior are not influencing the characteristics of drift we are observing in this work. To confirm that there is no significant difference in the behavior of mice across sessions, we trained an SVC to distinguish sessions F1 and F2 from the average behavioral metrics of the corresponding session. Specifically, we used 10-fold cross validation and further averaged over 10 shuffles of the data and found the SVC achieved chance accuracy, 47.3 ± 2.0% (mean ± s.e.). Similarly, for distinguishing N1 and N2, we found 48.9 ± 2.2% accuracy (mean ± s.e.).

To understand the differences in behavior across session, let b_I be one of the four numbers representing the behavior of session I, we define the normalized behavior difference as (16) and defined similarly for the novel session. For running speed, pupil area, and x-y eye position, we find the average normalized behavioral difference to be close to zero across all mice (S7(d) and S7(g) Fig). Finally, for each mouse, we can investigate if its change in behavior across sessions explains the size of the drift observed. For example, using 1d linear regression, the change in mean run speed across session does not help explain the amount of drift we observe (S7(e) and S7(h) Fig). Furthermore, fitting the drift to all four behavior metrics, we find the Hit and Miss trial drift we observe has p = 0.315 and 0.069, respectively for the familiar data, and for the novel data we find p = 0.201 and 0.261, respectively (4d linear regression, F-test).

4.3.2 Additional data quality control.

In order to verify that the behavioral data can reliably be compared across distinct imaging sessions, in this section we describe several additional quality control metrics that were performed on said data. We systematically show that several variances in day-to-day collection do not explain the drift characteristics we observe in the data. In particular, this section analyzes the region of interest (ROI) masks used to identify cells within a given session and also across sessions. Each cell in each session has an associated ROI mask that consists of binary values identifying where in the imaging plane luminescence was detected in the given session. More details about the ROI masks and the data processing that went into defining them can be found in the technical whitepaper [33].

To quantify the matching of a given ROI mask between two sessions, we introduce two metrics that can be computed for each cell that is matched across sessions. Let M^s(c) represent the number of nonzero values in the ROI mask of cell c in session s. To quantify how much the area of the ROI mask changes between sessions, we define the normalized area difference, A, of cell c to be (17) We will always take s to be the earlier session, i.e. s = F1 or N1. Note that A is positive (negative) when the later (earlier) session has more pixels.

In addition to area difference, we want a metric for how similar the shape of a cell’s ROI masks are between session. If we flatten the ROI masks into vectors, the normalized dot product between the resulting vectors can capture their similarity. However, we must compensate for the fact that the absolute X-Y location of a given cell in the imaging plane can change between the sessions. To this end, we scan the two masks over one another by iterating over a (pixel-valued) offset of (x, y) and seeing how this changes the alignment. Let be the flattened ROI mask of cell c in session s with an offset of (x, y). Define the maximum alignment, σ, to be (18) Note that we are only offsetting one mask from the other (the dot product would not change if we offset both masks the same amount). Since the components of a given m consist of only binary values, the maximum alignment obeys 0 ≤ σ ≤ 1.

With metrics to quantify how well each cell is matched between sessions, we investigated whether certain regions of the imaging plane were more poorly matched between sessions than others. This could occur if, for example, the imaging plane of the later session were tilted slightly relative to its location in the earlier session. To do so, we plotted the normalized area differences of cells as a function of their location in the imaging plane (S8(a) Fig). Across all mice and for both the familiar and novel drift, we then fitted the data using a 2d linear regression and used an F-test to determine that the fitted values did not differ significantly from a flat plane (S7(k) and S8(g) Figs). This same procedure was repeated for each cell’s maximum alignment between sessions (S8(b) Fig), also finding no significant evidence of a trend toward one direction in the plane (S8(h) and S8(l) Fig). We took this to mean, across all mice used in computing drift metrics, there was no significant trend of one particular region of the imaging planes being more poorly matched than other regions.

Next, we aimed to check if certain drift characteristics could be explained by how well cells were matched across sessions, as measured by the previously introduced measures. We plotted the amount of drift in a given cell (both in true value and magnitude) as a function of its normalized area difference (S8(c) and S8(d) Fig). Again, we fit this data using linear regression and found no significant evidence of drift varying as a function of this metric across mice and sessions (S8(i) and S8(m) Fig). A similar check as a function of maximum alignment also yielded no trend. Additionally, we plotted drift (again in true value and magnitude) as a function of the cell’s location in the imaging plane (S8(e) and S8(f) Fig). This was meant to investigate if any particular part of the imaging plane exhibited a statistically significant amount of drift. Analyzing the 2d linear regression fit, we again found no significant evidence of drift being biased toward one particular part of the plane (S8(j) and S8(n) Fig). Of course, absence of a significant result is not a conclusive proof of absence, but beyond the lack of significance, visually there does not appear to be a consistent relation between location in the imaging plane and cell matching, location in the imaging plane and drift, and quality of cell matching from session to session and the amount of drift.

We then checked if the quality of cell matching for all matched cells belonging to a given mouse could explain the amount of drift seen in said mouse across sessions. To do so, we plotted the magnitude of drift (normalized by the number of cells imaged) as a function of both the average magnitude of normalized area difference (S9(a) and S9(c) Fig) as well as the average alignment (S9(b) and S9(d) Fig). Fitting said data using linear regression, we again found no trend showing that the amount of drift observed, as measured by its magnitude, could be explained by the overall cell matching quality of the mouse.

Finally, we investigated if the qualitative results of drift geometry would change if we only kept cells that were “very-well-matched” between sessions. We defined a cell c to be well-matched if and . This criterion was quite strict and results in a large proportion of the matched cells being thrown out across mice (S9(e) and S9(i) Fig). Using the same pipeline we previously used to compute several important drift geometric characteristics, we find the very same qualitative properties are present in the drift when only the well-matched cells are included (S9(f)–S9(h) and S9(j)–S9(l) Fig).

4.3.3 Robustness of response vector.

To test the robustness of the mean response vectors used in this dataset, we a subset of trials and computed how much this caused the response vector to change. Let be the true mean response vector and let be the mean response vector once half of the members of the stimulus group are removed. We define the change in the mean response vector to be . For the novel data, removing either the even or odd Hit trials yields on average (for the familiar, ). Since we found the size of the drift to be comparable size of the mean response vectors (S2(e) and S2(f) Fig), this means the instability from removing half of the trials is small compared to the size of the drift vector. As expected, since there are significantly fewer Miss trials (generally 1/4 to 1/3 the amount compared to Hit trials, see Table 2) we find to be much higher, 0.43 for both the familiar and novel data.

Hyperparameter scan details.

To determine noise injection hyperparameters and generate the data shown in Fig 5a, we conducted scans over values of σ and p for each of the five types of noise. For node and weight dropout, we scanned over p values from 0.1 to 0.9 (in steps of 0.1). For additive node, additive gradient, and multiplicative type noise, we scanned over σ values of 10⁻³ to 10¹. Each network was trained until overfitting started to occur and the feature space representations from all 10 classes of CIFAR-10 were used to determine best fit metrics to the noise.

To evaluate the fit of our CNN models to the experimental data, we look across the four following criterion:

1. The percentage of drift’s magnitude that lies in variational space, and its constancy in Δt.
2. Drift’s trend of lying more and more obtuse from the largest PC dimensions, as a function of Δt. Note we condition this metric on the fact that this is well fit by a linear relationship, as it is in the experimental data, penalizing models whose r value is low even if the slope matches experiment well.
3. Drift’s tendency to lead to a flow of variance explained out of the top PC directions. Similar to the previous metric, we also condition this on a good linear fit.
4. Angle difference between SVC classifiers and the relative classification accuracy.

To quantify these four criterion, we evaluate the following sum of Z-score magnitudes (19) where Z(⋅) is the Z-score of ANN metric relative to the experimental data, , and are the Z-scores of the second and third metrics conditioned on good fits, (20) We use [⋅]⁰ to denote a clipping of the Z-score to be at most 0, so we do not penalize models that are better fits than the experimental data. Note the Z-score of the slope of the linear fit only contributes if its Z-score is smaller than that of the Z-score of the r-value.

We compute Z_total relative to the passive data, since that dataset has significantly more mice and thus sharper fits on all the parameters. Z-scores on the familiar/novel behavioral data are in general quite low and continue to be best fit by dropout-type noise, although the exact values of best fit for p and σ differ slightly.

Each of the four metrics contributes two Z-scores to this expression. Note we omit measuring model performance using metrics that vary significantly between the experimental datasets, for instance, the size of the drift magnitude relative means. We found that our overall conclusion, that node and weight dropout best match the experimental data, were not sensitive if we included the aforementioned metrics to evaluate performance.

Additional Fig 5 details.

With the exception of the hyperparameter scans, the rest of the ANN data in this figure was averaged over 10 separate initializations. Although training/testing were conducted over all 10 classes of CIFAR-10, plots here are for a single class, frogs. We found the qualitative characteristics of different classes to be the same, and the quantitative values did not vary significantly. Fig 5b and 5c were generated analogously to the Δt fits of Fig 2f and 2h, respectively (see above). Similarly, Fig 5d, 5e and 5f (as well as their node dropout analogs) were generated analogously to Fig 3a, 3b and 3c, without the averaging over stimulus groups.

Fig 6 details.

SVCs are trained with 10-fold cross validation using the same parameters used on the experimental data above, but now on the feature space representations of the two-class subset of CIFAR-10. Once again, the angle between SVCs is the angle between their normal vectors and cross classification accuracy is defined as in Eq (14) above. To produce Fig 6a, we computed the pairwise alignment between all SVCs within a window of 10 epochs and averaged this quantity over 10 initializations. Fig 6b used the same pairing but computed the cross classification accuracy, Eq (14), between all pairs. Note the time difference can be negative here because the data could be at a later time than when the classifier was trained. Fig 6c used the same pairing scheme as well, but instead computed the angle between the readout vectors for each class, and then averaged the quantities across classes. For Fig 6d, for a given class, we computed the stimulus group’s drift relative to the direction of its corresponding readout vector. The drift of a stimulus group compared to the other group’s readout looked similar. The chance percentage was computed by generating two random vectors in a space the same as as the feature space (n = 84) and computing their average deviation from perpendicular. Our previous findings indicate that the direction of drift is far from completely random in neural state space. However, they are not inconsistent with drift occurring in a high-dimensional space. The chance percentage shown in Fig 6d changes little if we restrict the drift direction to be constrained along many directions.

The targeted node dropout results in Fig 6e as well as in S4 Fig were generated by preferentially targeted particular nodes during dropout. Let P_μ for μ = 1, …, n be the probability of dropping a particular node in the feature layer during each forward pass. For the regular node dropout in this figure, which is identical to the node dropout data in Fig 5, we simply have P_μ = p = 0.5 for all μ = 1, …, n. To compute targeted dropout, 10 times per epoch we collected the feature space representations of the test set. Across the test set, we compute the variance for each node and use this to compute the ratio to total variance for each node, 0 ≤ v_μ ≤ 1 (this is similar to the ratio of variance explained of PC dimensions, v_i). Using this ratio of total variance for each node, 10 times per epoch we update the P_μ. For targeting nodes of maximum variance, we use the expression (21) where A ≥ 0 controls the strength of the dropout and clips the value between 0 and 1. For A = 1, on average only a single node is dropped, since ∑_μ v_μ = 1. Meanwhile, to target the nodes of minimum variance, we used (22) where ϵ = 10⁻⁵ sets a lower threshold for the smallest variance ratio. For the aforementioned figures, A = 20 and 42 for the maximum and minimum targeted node dropout data, respectively. This results in an average number of nodes dropped of ∼17 per forward pass for the maximum targeted nodes and 42 for the minimum node targeting (note the latter of these quantities is the same number that would be dropped under regular node dropout with p = 0.5, since pn = 42). Targeted node results were averaged over 5 initializations.

To generate the plot in Fig 6f, we lowered the frequency of the recalculation of what nodes are dropped to once per epoch (down from once for every forward pass). All other training parameters of the network are identical to the node dropout case in the main text. Since we sample the feature space 10 times per epoch, this means the sampling rate of the feature space is smaller than the noise update scale. This data was computed over 5 network initializations.