Representational drift reflects ongoing balancing of stochastic changes by Hebbian learning

Significance

Recent studies have shown that neural responses to familiar sounds continuously change in stable environments, a phenomenon termed representational drift. By analyzing long-term recordings from the mouse auditory cortex, we suggest that this drift results from an interplay between two processes: Hebbian-like plasticity, where neural connections are strengthened through coactivation, and stochastic synaptic changes that randomly alter neural connectivity. Using correlation analyses and computational modeling, we show that both processes are necessary to explain the observed patterns of neural activity. These findings provide mechanistic insights into how neural circuits maintain stable function despite continuous reorganization, with implications for understanding learning, memory, and neural adaptation.

Abstract

Recent evidence indicates that even under stable environmental and behavioral conditions, responses to sensory stimuli undergo continuous reformatting over the course of days, a condition described as representational drift. However, the processes underlying this phenomenon remain poorly understood. Examining the dynamics of signal and noise correlations among neuron pairs in chronic calcium imaging experiments in the mouse auditory cortex, we investigate how activity-dependent, Hebbian-like plasticity and activity-independent, stochastic synaptic processes contribute to representational drift. We found that signal correlations predict future noise correlations, suggesting that stimulus-induced coactivation leads to increased effective connectivity between neuron pairs. Moreover, simple linear network models were able to account for the observed temporal dependencies between signal and noise correlations, but only if both Hebbian-like plasticity and stochastic changes of either inputs or recurrent synapses contribute to representational drift. In conclusion, our findings suggest that continuous sensory input–driven Hebbian-like plasticity can balance ongoing stochastic synaptic changes, thereby preventing the network’s functional degradation.

Although the same sensory stimulus is generally thought to evoke a consistent and stable percept over time, studies show that the corresponding neuronal activity patterns can undergo substantial changes over the course of days, a phenomenon known as representational drift. In recent years, evidence for representational drift has been observed in various areas of the brain (1–5) suggesting that it reflects a fundamental property of neuronal circuits with possible implication for various cognitive functions (6). Despite its prevalence, the processes underlying representational drift remain largely unresolved.

Synaptic changes in network connectivity are a likely driver of changes in stimulus-evoked activity patterns (7). Traditionally, synaptic changes have been linked to activity-dependent Hebbian processes, where specific patterns of pre- and postsynaptic activity can modify the strength of synaptic connections between neurons. This form of plasticity is crucial for learning and memory formation (8), and recent modeling studies have suggested that Hebbian learning may also contribute to representational drift. Ongoing Hebbian plasticity may continuously modify the current network configuration (9), which could cause sensory representations to drift over time. This concept is akin to statistical learning, where organisms continuously adapt and potentially overwrite previously established memories with new information from constantly changing inputs (10). Other models explain drift by ongoing Hebbian learning in a degenerate solution space during noisy representation learning (11, 12). These modeling studies are in line with recent experimental findings, which suggest that not only experience but also time can influence the nature and degree of representational drift (13–15). These theoretical works have highlighted activity-dependent plasticity mechanisms as key contributors to drift, in combination with synaptic noise.

Indeed, synaptic connections exhibit substantial and seemingly stochastic fluctuations independent of neuronal activity, observed both in vitro and in vivo (16, 17). These ongoing changes in synapses were also observed under conditions where neuronal activity was silenced (18–20) possibly reflecting fluctuations in the dynamic molecular makeup of individual synapses (21). Thus, alongside Hebbian synaptic plasticity, activity-independent, stochastic changes may also influence the stability of activity patterns (7), although their role has been less studied to date. Furthermore, the relationship between these two forms of synaptic changes and their respective contributions to representational drift remains unclear.

Signal and noise correlations can provide a sensitive readout to changes in the activity and structure of a network. Foundational studies have explored the roles of synchronous activity and neuronal correlation in neural binding and visual processing (22, 23), as well as in the reliability and efficiency of neural coding (24–27). Many studies distinguish between signal correlations, which reflect the tuning similarity of neuron pairs (28, 29), and noise correlations, which capture the shared variability in neuronal responses independent of external stimuli. Studies on the origins of noise correlations suggest that they can depend on monosynaptic and multisynaptic connections, recurrent pathways, and stimulus independent shared inputs (24, 30, 31) suggesting their interpretation as effective connectivity between neurons (for a comprehensive review of both signal and noise correlations, see ref. 27). Despite their importance and broad applications, the long-term dynamics of both signal and noise correlations over days remain largely unexplored.

Importantly, signal and noise correlations may be affected differently by Hebbian-type, experience-dependent plasticity than by stochastic, experience-independent synaptic changes. Large signal correlations could, via Hebbian plasticity, promote the establishment of stronger synaptic connections between correlated neurons and thus predict changes toward enhanced noise correlations between such neurons. Conversely, stochastic synaptic drift may also affect both signal and noise correlations, but without any systematic temporal relationship between the two. Thus, changes in signal and noise correlations could potentially reveal important constraints on the plasticity processes underlying representational drift, but so far this possibility has been little explored.

In this study, we utilized signal and noise correlation to shed light on the relative contributions of the different forms of synaptic changes to representational drift. We analyzed chronic calcium imaging data of sound evoked responses in the mouse auditory cortex (5), a model well suited to address this question as both synapses (16, 32–34) and neuronal representations (5, 35–37) have been documented to undergo changes under stable environmental and behavioral conditions. We found that signal correlations predict future noise correlations, but not vice versa. The simplest network model that could account for these findings combined a Hebbian-like learning process in recurrent synaptic connections with a component of stochastic changes: either from drifting inputs or stochastic changes within the recurrent connections themselves. Furthermore, when examining the dynamics of signal and noise correlations during auditory cued fear conditioning, we observed a diminished influence of signal correlations on noise correlations. Together, these results suggest that Hebbian learning counterbalances ongoing stochastic changes during basal conditions, whereas the impact of stimulus-driven changes is slightly reduced after fear conditioning.

Results

To test the role of synaptic changes underlying representational drift, we analyzed our publicly available dataset (38) obtained using chronic two-photon calcium imaging in 80 fields of view in layers 2/3 of 10 awake, head-fixed mice. Neurons were expressing the calcium indicator GCaMP6 together with a structural nuclear marker enabling high-fidelity tracking of individual neurons across several days. Animals were passively listening to 20 trials of 34 short stimuli (50 to 70 ms) in randomized order in four imaging sessions separated by two days each. Cells were tracked across all imaging sessions and neuronal recordings were preprocessed resulting in activity traces for each of the 18,249 neurons (for a detailed description of recording procedures and data handling, see ref. 5 and Materials and Methods). Between the second and third imaging session (i.e. days 3 and 5), the mice underwent a fear conditioning paradigm on day 4, where one of the stimuli was paired with a mild electric shock (Fig. 1A). An example field of view is shown in Fig. 1B. Neuronal somata are labeled in red and GCaMP6m expression is shown in green. Example activity traces for two cells can be seen in Fig. 1C, both single trial activity in black and median activity across trials in purple (for further examples, see SI Appendix, Fig. S1).

Fig. 1.

Experimental data and estimation of signal and noise correlations. (A) Experimental timeline. (B) In vivo two-photon image of a local population of neurons in layer 2/3 of the auditory cortex showing expression of GCaMP6m (green) and H2B::mCherry (red) (for details, see ref. 5). Arrows indicate location of the neuron pair shown in (C, E, F, H, and I). White circles show location of the neurons shown in (J and K). Neurons were selected for high signal correlation, noise correlation, or activity. (C) Single trial activity (black line) and median stimulus-specific response over trials (purple line) of the example neuron pair with particularly high signal and noise correlation shown marked by arrows in (B). The green box indicates the stimulus responses displayed in (H). Gray and white columns indicate stimuli. (D) Sketch illustrating signal correlations as a measure of tuning similarity. (E) Median stimulus-specific response of the example neuron pair in (C). (F) Median stimulus-specific response of neuron 126 vs. neuron 127. (G) Sketch of the interpretation of the noise correlation as a measure of an effective connectivity between neurons. (H) Single trial activity of the example neuron pair in (C) in response to stimulus 3 [Zoom-in on the green boxes shown in (C)]. The purple line represents median response over trials. (I) Trial-to-trial variability for all trials and stimuli of neuron 126 vs. neuron 127. (J) Activity traces of 50 example neurons which have a maximum stimulus response of $>$0.4 and a maximum signal or noise correlation of $>$0.75 [marked by white circles in (B)]. Neurons are sorted by hierarchically clustering the signal correlation matrix, trials are sorted by stimuli. (K) Signal and noise correlation matrix of the neurons shown in (J). Lower triangular matrix shows pairwise noise correlations, upper triangular matrix shows pairwise signal correlations.

We analyzed signal correlations, reflecting the stimulus-driven input to the cortical network (Fig. 1D–F) and noise correlations which served as a proxy for effective cortical interactions (Fig. 1G–I). Signal correlations between pairs of neurons were obtained by first computing each neuron’s median response across trials for each stimulus (Fig. 1E) and then calculating the Pearson correlation coefficient between median stimulus-specific responses for each neuron pair (Fig. 1F). To compute noise correlations, we first subtracted for each stimulus the median activity from each single trial response (Fig. 1H). Computing the Pearson correlation coefficients of the residual activity between all pairs of neurons then yielded the noise correlations (Fig. 1I). To obtain a more robust estimate of both signal and noise correlations, we cross-validated our estimates via bootstrapping (see Materials and Methods for details). Fig. 1J shows for a representative subset of cells from an example field of view their stimulus-evoked activity and Fig. 1K their corresponding pairwise signal and noise correlations. Note that high signal correlations between pairs of neurons at a given imaging time point do not necessarily indicate high noise correlations and vice versa.

A Dynamic Equilibrium of Signal and Noise Correlations During Representational Drift.

To explore the dynamics of signal and noise correlations, we first plotted their distributions at a given day, and observed that both display a mode near zero and a tail extending toward positive correlations (signal correlations: Fig. 2A, noise correlations: Fig. 2D), consistent with previous reports (39, 40). Both distributions were remarkably stable across all four imaging time points. Next, we investigated whether correlations were stable at the level of individual neuron pairs. To assess the stability of signal correlations, we first divided the range of correlation values into eight bins and assigned neuron pairs into these bins based on their signal correlation on the first imaging day. We then plotted the signal correlation distribution for each group separately on the next imaging day (day 3), while maintaining the group assignment from day 1. This revealed that signal correlations varied substantially across days and for each group correlations tended toward the original overall distribution (Fig. 2B). Plotting the mean signal correlation for each group at both time points illustrates the drift of these means toward a common mean (Fig. 2C). Moreover, applying an analogous procedure to noise correlations revealed that these show similarly strong changes as signal correlations (Fig. 2 E and F). Note that these changes across days are unlikely to be caused by experimental noise, as we estimated the noise level through cross-validation (Materials and Methods). Taken together, both signal and noise correlations were highly volatile under basal conditions across a time span of two days, while their overall distributions were maintained.

Fig. 2.

Signal and noise correlations are in a highly volatile state. (A) Distribution of pairwise signal correlations across days. (B) Left: Neuron pairs were assigned to groups based on their signal correlations on day 1. Color indicates threshold used for grouping (see legend). Probability normalized over all neuron pairs. Right: Signal correlation distribution of each group on day 3. Probability normalized over neuron pairs within groups. (C) Mean signal correlation of each group on day 1 and day 3. Error bars represent SEM across neuron pairs. (D–F) Same as (A–C) but for noise correlations.

Signal Correlations Predict Future Noise Correlations.

Having established that signal and noise correlations show substantial changes over time, we next wondered whether some of these changes were consistent with ongoing Hebbian learning. To address this question, we examined the relation between the changes in signal correlation and noise correlation. As Hebbian learning posits that two neurons strengthen their connectivity if they are coactivated, we hypothesized that neuron pairs coactivated by the same stimuli may exhibit stronger connectivity at a later time point due to Hebbian learning, which would tend to strengthen the noise correlation in such pairs. In this case, neuron pairs with high signal correlations should have increased noise correlations on the next imaging day.

To test this hypothesis, we investigated the extent to which the signal correlation levels of neuron pairs predicted their noise correlations on the next imaging day (Fig. 3A). To minimize the factor of noise correlation levels on day 1 in this analysis, we began by grouping neuron pairs based on their noise correlations, as we had done previously (Fig. 2F). In Fig. 3B we illustrate our analysis using the highest noise correlation group (NC $=$ 0.6 to 1; Fig. 3B, Left). First, we subdivided this group into eight subgroups according to their signal correlation on day 1 (Fig. 3B, Right). Given the low correlation of signal and noise correlations on a given day (Fig. 1K and SI Appendix, Fig. S2), neuron pairs in this group are expected to cover a wide range of signal correlations. For each signal correlation subgroup, we then computed the mean noise correlation on day 1 and on day 3 (i.e. the subsequent imaging day) and found that subgroups with high signal correlations on day 1 exhibited systematically higher noise correlations on day 3 than those with low signal correlations (Fig. 3B, Middle). At the same time, noise correlations differed only little on day 1, indicating that this effect cannot be explained by differences in noise correlation on day 1. We applied the same analysis to the other noise correlation groups and found that such temporal dependency between signal and noise correlations across days was also evident for the lower noise correlation groups albeit to a lesser degree (the three highest NC groups are depicted in Fig. 3C; all groups are shown in SI Appendix, Fig. S3A). These results indicate that cell pairs with a high noise correlation on day 1 tend to exhibit less decay in noise correlation from day 1 to day 3 if their signal correlation is high on day 1. Moreover, using the same analysis to assess the influence of noise correlations on future signal correlations revealed a weaker predictive effect (Fig. 3 D and E). Although the subgroups with high noise correlation tended to have a slightly increased signal correlation on the next day, the differences in mean signal correlation across subgroups were low (Fig. 3F, for a complete overview of all subgroups, see SI Appendix, Fig. S3D).

Fig. 3.

Signal correlations predict noise correlations under basal conditions. (A–C) Assessment of the predictive effect of signal correlations on noise correlations. (A) Illustration of this effect. (B) Left: mean noise correlation on days 1 and 3 of neuron pairs with noise correlation 0.6 to 1 on day 1. Right: mean noise correlations on days 1 and 3 of the same neuron pairs as on the Left, but subgrouped by their signal correlation. Individual lines represent individual signal correlation subgroups, color indicates signal correlation on day 1. (C) Same as in (B), but for two more noise correlation groups. The Right subplot corresponds to the group shown in (B). All error bars in SEM. (D–F) Assessment of the predictive effect of noise correlations on signal correlations. Same as (A–C) but with reversed signal and noise correlations. (G) Illustration of the prediction index (P.I.). It quantifies the predictive effect of signal correlations on noise correlations, or vice versa, by assessing the ordering and the spread of the lines due to the subgrouping. For illustration, the prediction index is indicated for four toy examples. (H) Prediction index of signal correlation day 1 on noise correlation day 3 for the different subgroups defined in (B) (orange line, mean $\pm$ SD of bootstrapped distribution). Shuffle in gray (mean $\pm$ SD). (I) Prediction index of noise correlation day 1 on signal correlation day 3 (blue line, mean $\pm$ SD of bootstrapped distribution). Shuffle in gray (mean $\pm$ SD). *$P<0.05$ (WSR test).

To quantify these effects, we defined the prediction index (Fig. 3G; see Materials and Methods). The prediction index assumes a maximal value of 2 if the mean correlations on day 3 are maximally spread apart and sorted in the same order as the subgroups on day 1 (Fig. 3G, Left, illustrates this for the case of noise correlations and subgroups based on signal correlations on the first day (coded by gray values). The prediction index reaches its minimal value of $-$2 if the spread is maximal in reverse order (Fig. 3G, Middle Left), while its value is 0 if there is no ordered spread (Fig. 3G, Middle Right) or no spread at all (Fig. 3G, Right). Thus, the prediction index quantifies the extent to which the order of subgroups on day 1 corresponds to the spread of correlations on day 3, offering a metric for assessing the predictive power of signal correlations on future noise correlations and vice versa. Calculating this prediction index for each group, we found that under basal conditions, signal correlations have a significantly predictive effect on future noise correlations (Fig. 3H). To assess the significance of our findings, we compared them to a shuffled control, where neuron pairs were randomly reassigned to correlation groups, thereby disrupting the structured relationship between signal and noise correlations while preserving their marginal distributions. The predictive effect of noise correlations on future signal correlations was significant, too, but substantially weaker than that of signal correlations on future noise correlations (Fig. 3I).

Signal Correlations Stabilize Noise Correlations.

Consistent with this asymmetric predictive relationship between signal and noise correlations (Fig. 3), we also observed a stabilizing effect using an alternative metric focused on correlation stability rather than prediction. Specifically, we grouped neuron pairs based on their signal correlation on day 1 and quantified the stability of noise correlations separately for each group by calculating the Spearman rank correlation of noise correlations between day 1 and day 3 (Fig. 4A–C, for all signal correlation bins, see SI Appendix, Fig. S4A). This analysis revealed an almost linear increase in noise correlation stability between day 1 and day 3 with increasing signal correlation on day 1 (Fig. 4B). A linear fit confirmed this trend, and the effect disappeared when neuron pairs were shuffled across signal correlation bins (Fig. 4C). Conversely, grouping by noise correlation and evaluating the stability of signal correlations did not reveal a similar effect. Signal correlations were generally more stable than noise correlations, but this stability did not systematically depend on the level of noise correlation (Fig. 4D–F, for all noise correlation bins, see SI Appendix, Fig. S4D). These results, thus, corroborate our findings in Fig. 3 and complement them by highlighting the stabilizing role of signal correlations on noise correlations.

Fig. 4.

Signal correlation stabilizes noise correlation under basal conditions. (A) Noise correlation on day 1 vs. noise correlation on day 3 for groups of neuron pairs with different signal correlation on day 1. Shown are 4 out of 8 groups. (B) Noise correlation stability measured as the Spearman rank correlation between noise correlation on day 1 and day 3 for each signal correlation group (mean $\pm$ SD of bootstrapped distribution). Shuffle in gray. (C) Slope obtained from linear fit of noise correlation stability to signal correlation on day t. (D–F) Same as (A–C), but with switched signal and noise correlations. *$P<0.05$ (MWU-test).

This stabilizing effect of signal correlations on noise correlations can be interpreted as a consequence of Hebbian plasticity: neuron pairs with similar stimulus tuning (high signal correlation) are coactivated and therefore strengthen existing functional connections between these neurons, resulting in more stable effective connectivity, as measured by noise correlations.

Fear Conditioning Reduces the Stabilizing Effect of Signal Correlations on Noise Correlations.

To examine how behavioral learning impacts the relationship between signal and noise correlations, we analyzed the effect of auditory cued fear conditioning (ACFC), using our previously published data (38), where one stimulus was paired with a mild electric shock between days 3 and 5. While fear conditioning did not significantly affect the prediction indices (SI Appendix, Fig. S5), we observed slight changes in the correlation stability relationships. The stabilizing effect of signal correlations on noise correlations was significantly reduced after fear conditioning, as evidenced by a reduction of stability for high signal correlation immediately following fear conditioning and a decreased slope in the linear relationship between signal correlation strength and noise correlation stability (Fig. 5 A and B). Conversely, noise correlations gained increased influence over signal correlation stability following fear conditioning (Fig. 5 C and D). These rather counterintuitive modulations may facilitate memory consolidation by limiting ongoing stimulus-induced plasticity.

Fig. 5.

Fear conditioning reduces the stabilizing effect of signal correlations on future noise correlations. (A) Spearman rank correlation between noise correlation on imaging days t and t$+$1 increases with the signal correlation on day t. (B) Slope from linear fit of noise correlation stability to signal correlation on day t. (C) Spearman rank correlation between signal correlation on day t and day t$+$1 as a function of noise correlation on day t. (D) Slope from linear fit of signal correlation stability to noise correlation on day t. Error bars represent SD of bootstrapped distribution. n.s. nonsignificant, *$P<0.05$ (MWU-test).

Together, our empirical findings show that under basal conditions, signal correlations are predictive of future noise correlations, whereas noise correlations are barely predictive of future signal correlations. Moreover, high signal correlations appear to stabilize noise correlations across days. Behavioral learning does not significantly alter this predictive relationship and only slightly modulates the stabilizing effect. Such temporal relationship appears consistent with ongoing Hebbian plasticity induced by sensory input, which may thus be a strong factor underlying representational drift. To disentangle the contribution of Hebbian plasticity from stochastic, input-independent processes, we next study in a network model possible underlying mechanisms for such a temporal relationship between signal and noise correlations.

Minimal Circuit Models Suggest Interplay of Hebbian-Like and Stochastic Plasticity.

To explore potential mechanisms underlying the observed dynamics in signal and noise correlation, including their asymmetric temporal relationship, we employed a linear recurrent network model (Fig. 6A). This model was designed to identify the minimal plasticity mechanisms that could explain our experimental results. The dynamics of the network are described by the following equation:

$$\begin{array}{c}\hfill \frac{\text{d}{r}_i}{\text{d}t}=-r_i+{\displaystyle \sum _{j=1}^N}{M}_{\mathit{ij}}{r}_j+s_i+{\xi }_i,\end{array}$$ [1]

Fig. 6.

Circuit models suggest an interplay of Hebbian-like and stochastic plasticity is required to account for the experimental observations. (A) Sketch of model architecture. (B) Experimentally observed mean prediction index of signal correlations (SC) on noise correlations (NC) (Left bar) and of noise correlations on signal correlations (Right bar), summarizing Fig. 3. Bar color reflects mean prediction index using the same color bar as in (C–I). (C–E) Models with stochastic changes. Mean prediction index in a model with stochastic input drift (C), stochastic network drift (D), and stochastic input and network drift (E). Left: sketch of network mechanism. Middle: mean prediction index of signal correlations on noise correlations. Right: mean prediction index of noise correlations on signal correlations. (F) Model with Hebbian-like mechanism. Top: sketch of network mechanism. Middle: mean prediction index of signal correlations on noise correlations. Bottom: mean prediction index of noise correlations on signal correlations. (G–I) Model with stochastic and Hebbian-like mechanism. Mean prediction index in a model with stochastic input drift and network plasticity (G), stochastic drift and plasticity in the network weights (H), and all three mechanisms combined (I). Left: sketch of network mechanism. Middle: mean prediction index of signal correlations on noise correlations. Right: mean prediction index of noise correlations on signal correlations. Parameter scan in (I) shows results for one example net-work drift strength (${\omega }_{\text{netw}}=0.4$). Black contours indicate plausible parameter spaces in agreement with data.

where $r_i$ is the firing rate neuron $i$, $M_{\mathit{ij}}$ represents the synaptic weight from neuron $j$ to neuron $i$, $s_i$ is the stimulus part of feedforward input to neuron $i$, ${\xi }_i$ is neuron-specific additive Gaussian noise, and $N$ is the total number of neurons in the network. To generate a dataset similar to experimental data we applied matching experimental parameters such that the same analysis procedures could be applied (Materials and Methods and SI Appendix, Fig. S6 A–H).

In this model, we explored which plasticity mechanisms could account for the following key experimental observations: the high volatility of signal and noise correlations over time (Fig. 2B, C, E, and F), the predictive effect of signal correlations on future noise correlations (Fig. 3 H and I), and the stabilizing role of signal correlations on noise correlations (Fig. 4 B and C). We assumed that synaptic weight changes in the network can occur through activity-dependent processes, such as Hebbian plasticity, or via stochastic activity-independent processes. We focused on three potential mechanisms, either in isolation or combined: A) stochastic changes in the input, B) stochastic changes in the recurrent network weights, and C) input-based “Hebbian-like” plasticity in the recurrent connections (see Materials and Methods for definitions).

We first tested whether stochastic changes alone could account for the observed dynamics of signal and noise correlations, considering changes only in the inputs, changes only in the recurrent connections, or a combination of both. If such stochastic changes alone were sufficient to account for our empirical observations, then this would suggest the possibility that these could be explained without the need for activity-dependent plasticity mechanisms like Hebbian learning. To test this hypothesis, we implemented stochastic changes for both inputs and recurrent connections in the model using an Ornstein–Uhlenbeck process (41) to introduce variability while maintaining overall statistics in a steady state. We varied the rate of change and computed the prediction index across subsequent time points (Materials and Methods and SI Appendix, Fig. S6 I–N) to test for temporal dependencies between signal and noise correlation, and vice versa (Fig. 6). In addition, we performed the conditional correlation stability analysis (SI Appendix, Fig. S8) analogously to the experimental data (Fig. 4 and SI Appendix, Fig. S4).

First, we examined the effect of stochastic changes in the input $s_i$ (Fig. 6C, Left). This allowed signal correlations to fluctuate significantly, mirroring the volatility observed in experimental data. However, while such stochastic changes in the input could account for the changes in signal correlations, they failed to capture the changes in noise correlations, simply because noise correlations depend only on the recurrent connectivity in our model (SI Appendix, Fig. S6G). Importantly, irrespective of the rate of synaptic changes, noise correlations depended little on signal correlations, with a prediction index close to zero (Fig. 6C, Middle), indicating that input drift alone cannot explain the observed interaction between signal and noise correlations. Next, we tested stochastic drift in the recurrent synaptic weights $M_{\mathit{ij}}$ while keeping the input constant (Fig. 6D). Again, whereas such drift in the recurrent connections could account for the overall volatility, the predictive effect of signal correlations on noise correlations remained absent (Fig. 6D, Middle). Finally, we combined stochastic changes in both the input and recurrent connections. As expected, this combination produced volatility in both signal and noise correlations, similar to what we observed experimentally, but still failed to reproduce the predictive effect of signal correlations on noise correlations, as the prediction index remained near zero (Fig. 6E, Middle). Moreover, for large combinations of drift rates, the model produced considerably positive or negative prediction indices for noise correlations on signal correlations (Fig. 6E, Right). These effects are primarily caused by strong recurrent connections, but are inconsistent with our experimental observations (Fig. 6B). Additionally, at higher levels of recurrent synaptic drift, we observed a reverse ordering of the correlations, likely due to the mean-reverting nature of the stochastic process, leading to extreme values being indicative of a smaller value at a consecutive time point and thus resulting in a negative prediction index (Fig. 6D). Thus, while stochastic processes, specifically in recurrent connections, are able to account for the high volatility of signal and noise correlations observed in the experimental data, they are insufficient to explain the predictive relationship between signal and noise correlations.

Given that stochastic processes alone could not explain the experimental data, we next implemented a simple Hebbian-like learning mechanism to investigate whether input-driven plasticity could reproduce the observed dynamics of signal and noise correlations. Hebbian-like learning was implemented as an interpolation between the connectivity matrix at the previous time step and a term reflecting the structure of the input vectors (akin to the learning rule in a Hopfield model; see Materials and Methods). Using this learning mechanism alone, the model was indeed able to reproduce the predictive effect of signal correlations on noise correlations (Fig. 6F). However, after several iterations, the network weights converged to the values imposed by the external input, causing both signal and noise correlations to stabilize and lose their volatility.

Anticipating that stochastic changes could possibly counteract this restrictive tendency of such a pure Hebbian-like learning mechanism (42), we next explored whether both processes together could explain the experimental results, by combining the above-described stochastic changes in the input and Hebbian-like plasticity in the recurrent network (see Materials and Methods for details). This combination successfully reproduced both key features: the ongoing volatility in signal and noise correlations, as well as the predictive effect of signal correlations on noise correlations (Fig. 6G). The input drift introduced sufficient variability in the external stimuli, thus preventing the network from converging, while the Hebbian-like plasticity ensured that coactive neurons developed stronger connections, maintaining the predictive relationship between signal and noise correlations. We also examined a model combining stochastic changes in recurrent connections with Hebbian-like plasticity. We find that this combination also successfully reproduces both the volatility of signal and noise correlations and the predictive effect of signal correlations on noise correlations (Fig. 6H). In this model, the stochastic drift in the network drives continuous synaptic changes, while Hebbian-like plasticity prevents the network from becoming overly disorganized. Furthermore, the combination of all three mechanisms, i.e. input drift, network drift, and Hebbian-like plasticity, also successfully reproduces the experimental observations (Fig. 6I and SI Appendix, Fig. S7). Note that while our model accounts for the mean prediction indices, it fails to capture the exact relationship between noise correlation strength and prediction index magnitude (compare Fig. 3H and SI Appendix, Fig. S6N), potentially due to the overly simplified connectivity structure in the model (see Discussion for details).

Moreover, when combining stochastic and Hebbian-like plasticity, the model also accounted for the experimental stabilization effects shown in Fig. 4. Specifically, using conditional correlation stability analysis, analogous to the analysis of the experimental data (Fig. 4), the model reproduced the asymmetric relationship in which signal correlations stabilize noise correlations, but not vice versa—however, only when Hebbian-like plasticity was combined with at least one stochastic process, either input drift, network drift, or both (SI Appendix, Fig. S8).

To understand how recurrent connection strength influences these dynamics, we systematically varied the net connectivity strength (SI Appendix) in our model, while keeping other parameters unchanged. We found that the predictive and stabilizing effect of signal correlations on noise correlations emerges as long as the net connectivity is moderately strong and the effect tends to become more pronounced with increasing connectivity strength (SI Appendix, Fig. S9 A, B, D, and E). For too weak connections feedforward inputs dominate the network dynamics, preventing recurrent connections from mediating the relationship between signal and noise correlations. Conversely, in the limit of strong connectivity, the network dynamics approaches the edge of instability (maximal eigenvalue of connectivity matrix equal to 1). The recurrent connectivity strength used in Fig. 6 (maximal eigenvalue of 0.85) is representative of a broad regime in which recurrent connectivity is strong enough to have an influence, while the dynamics is still far away from the edge of instability.

To address whether our findings generalize to more biologically plausible network models, we also tested our conclusions in a balanced, nonlinear model (SI Appendix). Varying again systematically the connectivity strength, we found that when combining input drift and Hebbian-like network plasticity, the prediction index reproduced experimental results if the connectivity was sufficiently strong (SI Appendix, Fig. S9C). The same held true, albeit slightly less pronounced, for network drift combined with Hebbian-like network plasticity (SI Appendix, Fig. S9F). These results indicate that our conclusions regarding the interplay between Hebbian-like learning and stochastic processes underlying representational drift also hold for more realistic types of network models.

Finally, to delineate more precisely which aspects of Hebbian plasticity are required to explain our experimental data, we reimplemented the Hebbian part of the plasticity mechanism, by now driving synaptic updates using the steady state network responses to input stimuli. This scheme and the input-based update scheme used throughout the rest of this study can be viewed as two opposite approximations to an ongoing, activity-based Hebbian learning rule: the former emphasizing the network-iterated responses after sustained stimulation, the latter the initial responses briefly after stimulus onset that reflect most strongly the feedforward input. Therefore, comparing the effects of both with the data could allow us to reveal which of these two regimes might be more important for the effects reported. Thus, we repeated the prediction analysis as described above exchanging in the Hebb term the input with the steady-state response vectors (SI Appendix), focusing on the model with input drift $+$ network plasticity, network drift $+$ network plasticity, for both low and high values of net connectivity strength. Notably, across all tested parameter combinations, this plasticity rule failed to reproduce the experimental observations: the model no longer showed a predictive relationship between signal and noise correlations (SI Appendix, Fig. S9 G–J). For low connectivity strength, stimulus inputs dominated the activity, and the recurrent network structure had little effect, resulting in no predictive or stabilizing effect. For stronger connectivity, the network structure dominated the response, causing the plasticity to reinforce internal network response modes rather than stimulus-specific coactivation patterns. These results suggest that Hebbian learning driven primarily by the feedforward inputs, rather than the sustained (steady-state) network responses is necessary to reproduce the temporal relationship between signal and noise correlations observed in the data.

In conclusion, within the framework of our recurrent network model, a balance between stochastic changes (either in the feedforward inputs or the recurrent connections) and input-based Hebbian-like learning in the recurrent connections was required to reproduce the temporal relationship between signal and noise correlations we observed in our experimental data.

Discussion

In this study, we investigated the mechanisms underlying representational drift in the mouse auditory cortex utilizing signal and noise correlations. Under stable environmental and behavioral conditions, we observed that both signal and noise correlations were highly volatile. Interestingly, signal correlations consistently predicted and stabilized noise correlations. Studying the conditions under which these observations can be accounted for by a simple network model, we interpreted this temporal pattern between signal and noise correlations as resulting from Hebbian-like plasticity balancing an ongoing stochastic process in the inputs to and/or the recurrent connections within the layer 2/3 cortical network.

Signal and noise correlation have been studied extensively and shown to be an important characteristic of the functional properties of neural networks (e.g. refs. 24, 30, and 31). Despite their importance in neural computation, relatively little is known about their long-term dynamics. Our findings showed that both signal and noise correlations are highly volatile under stable environmental and behavioral conditions. While this volatility might initially seem surprising, it is consistent with the well-documented perpetual remodeling of neuronal tuning properties termed representational drift (1–5). This drift is likely resulting from changes in the underlying synaptic connectivity in the network (16–18), that can be observed even in the absence of neural activity (18, 19), suggesting it reflects an intrinsic property of synapses (20).

Importantly, the volatility in correlations we observe was not entirely random, but the predictive and stabilizing effect of signal correlations on noise correlations suggests the involvement of a Hebbian process in representational drift. In fact, our model shows that Hebbian-like plasticity in recurrent synaptic connections when combined with stochastic changes in connectivity can explain this predictive and stabilizing effect of signal on noise correlations (Fig. 6G–I). Although our empirical observation of the temporal relationship between signal and noise correlations itself is only correlational, our model results suggest a causal link between the two, mediated by Hebbian-like plasticity, by showing that such plasticity can account for the temporal pattern we observed in the experimental data. Such a Hebbian-like mechanism offers a simple, parsimonious, and plausible explanation for our empirical observation, but a possible involvement of other processes, such as homeostatic plasticity, changes in intrinsic excitability, or more complex forms of synaptic plasticity, cannot be excluded.

Noise correlations are sometimes interpreted as an effective connectivity, and used to infer network structure, but often noise correlations provide only an indirect and imperfect proxy to synaptic connectivity. Our modeling framework suggests that the changes in noise correlations we observe that are predicted and stabilized by signal correlations reflect, at least in parts, changes in the underlying structural connections shaped by stimulus-driven coactivation. However, owing to the simplicity of our network model, it cannot be directly used to infer synaptic connectivity patterns. While our model captures the key experimental finding that signal correlations predict future noise correlations, it does not fully reproduce the observed relationship between noise correlation magnitude and prediction strength (compare Fig. 3H and SI Appendix, Fig. S6N). This discrepancy likely arises from architectural differences between the simplified model and cortical circuits. Specifically, the model assumes all-to-all connectivity, whereas cortical networks are sparsely connected, where only a fraction of neuron pairs are directly connected. Assuming Hebbian plasticity can only act on existing synapses and that noise correlations tend to be higher if neurons are directly connected, this may explain why, in the experimental data, stronger noise correlations are associated with higher prediction indices, a pattern not captured by our simplified model.

The stabilizing effect of signal correlations on noise correlations (Fig. 4) can likewise be interpreted as a signature of Hebbian-like plasticity, specifically Hebbian plasticity acting selectively on existing synapses as opposed to noise correlations induced by multisynaptic connections or shared inputs. Hebbian mechanisms would then strengthen these existing connections in proportion to their stimulus-driven coactivation, leading to enhanced stability over time. Despite reproducing the predictive and stabilizing effects apart from the above mentioned discrepancy (Fig. 3H), our model does not allow to resolve differences between direct and indirect connections due to its all-to-all connectivity.

Notably, the predictive and stabilizing relationship between signal and noise correlations in the model proved robust across a range of architectures and parameter regimes. In particular, the effect emerged in both linear and nonlinear networks, provided that the recurrent connectivity was sufficiently strong. This shows that this effect is robust and does not depend on a narrow set of assumptions and suggests that it may generalize to biologically more detailed models as well. Interestingly, however, when we tested a Hebbian-like plasticity rule emphasizing the sustained (steady-state) network responses, as opposed to being more directly shaped by feedforward inputs, the model consistently failed to reproduce the experimental findings. This suggests that biological networks may use specific additional mechanisms to ensure feedforward-driven plasticity, and this could involve gating of plasticity by sensory input, dendritic compartmentalization, or neuromodulatory control that selectively reinforces stimulus-evoked coactivation patterns. While our model does not resolve which mechanism could be at work, it suggests that Hebbian plasticity rules primarily depending on the initial response to feedforward-input contribute to representational drift.

Our findings build upon and extend recent theoretical work that has proposed Hebbian learning combined with synaptic noise as a mechanism for representational drift (9, 11, 12). We introduce the dynamics of signal and noise correlations as empirical readouts that enable direct testing of Hebbian mechanisms in chronic datasets. By directly linking these longitudinal data with such a mechanistic modeling framework, we provide empirical evidence for the idea, put forward by these models, that activity-dependent plasticity is an important component of representational drift. Our empirical and theoretical analyses together suggest that drift arises from the interplay between structured Hebbian-like learning and unstructured synaptic variability. Our pairwise correlation framework thus offers an experimental validation of the theoretical predictions made by these earlier models, and suggests a balanced role for stochastic and activity-dependent processes.

Assuming that the balance of two antagonistic processes controls representational drift, we hypothesize that this balance may be shifted by internal and external processes leading to higher or lower drift rates. Indeed, increased exposure to external stimuli in an enriched environment can slow down drift (4, 43). This could be interpreted in a way that augmented reinforcement of consistent sensory inputs reduces the relative impact of inherent stochastic synaptic changes. Thus, Hebbian mechanisms act to counterbalance this synaptic drift, with the input acting as a “stabilizing force,” preventing excessive disorganization that would result from stochastic changes alone. Such interpretation is in line with studies showing that Hebbian-like synaptic plasticity can stabilize network dynamics, ensuring functionally relevant connections to persist despite random synaptic turnover (11, 12, 44). Future experiments isolating and manipulating individual components of inherent stochastic synaptic dynamics or forms of Hebbian plasticity are needed to determine how input-driven and network-driven processes interact to shape representational drift, shedding light on the mechanisms that support ongoing learning and adaptation.

The two suggested underlying causes of representational drift, input-driven changes and intrinsic synaptic processes, bear interesting links to mechanisms of forgetting. In fact, theories of forgetting, such as trace decay and interference theory, offer parallels to the two primary processes we observe. Trace decay suggests that memories fade over time due to intrinsic changes, consistent with synaptic drift (45–47), while interference theory posits that new information replaces old, aligning with the idea of continuous incorporation of new sensory inputs (48, 49). A recent study in the mouse hippocampus supports both mechanisms, showing that drift in neuronal firing rates changes with time, while tuning properties of neurons change with experience (14). The link between drift and forgetting has also been made in a recent review (50), and our findings are consistent with the notion that representational drift could be an ongoing process of both intrinsic decay and constant extrinsic overwriting, reflecting both aspects of forgetting.

Together, our findings highlight that data acquired from a mature and apparently stable brain on a given day, merely reflect a snapshot of two counterbalancing dynamic processes safeguarding functionality in an inherently unstable network.

Materials and Methods

Overview.

A detailed description of all procedures is provided in SI Appendix. Below we summarize the key steps.

Experimental Data.

We analyzed publicly available chronic two-photon calcium imaging dataset from the auditory cortex of awake, head-fixed mice (38). Neurons expressing GCaMP6 and a nuclear marker were recorded across four imaging sessions separated by two days. Mice were passively exposed to 34 short auditory stimuli (pure tones and complex sounds, 20 repetitions each). Between sessions two and three, auditory cued fear conditioning was performed on day 4, pairing one stimulus with a mild foot shock. Data preprocessing followed published procedures (5, 51).

Signal Correlation (Figs. 1D–F, J, and K, Fig. 2A–C, Fig. 3A–F, H, and I, Fig. 4, Fig. 5, and Fig. 6B–I and SI Appendix, Figs. S1–S9).

Signal correlations (SC) were computed as the Pearson correlation of the median stimulus-specific responses ${\mathrm{\Gamma }}_i(s)$, ${\mathrm{\Gamma }}_j(s)$ for pairs of neurons $i$, $j$: ${\text{SC}}_{\mathit{ij}}=\rho \left({\mathrm{\Gamma }}_i,{\mathrm{\Gamma }}_j\right)$. Bootstrapping was used to obtain robust estimates.

Noise Correlation (Figs. 1G–K, Fig. 2D–F, Fig. 3A–F, H, and I, Fig. 4, Fig. 5, and Fig. 6B–I and SI Appendix, Figs. S1–S9).

Noise correlations (NC) were computed as the Pearson correlation of the residual trial-to-trial variability after subtracting the median stimulus-specific responses ${\mathrm{\Gamma }}_i(s)$, ${\mathrm{\Gamma }}_j(s)$ for pairs of neurons $i$, $j$: ${\text{NC}}_{\mathit{ij}}=\rho \left(X_i(s,T)-{\mathrm{\Gamma }}_i(s),X_j(s,T)-{\mathrm{\Gamma }}_j(s)\right)$, where $X_i(s,T)$ are neuronal activities of neuron $i$ for stimulus $s$ in trial $T$. Bootstrapping was used to obtain robust estimates.

Correlation Stability (Fig. 2 and SI Appendix, Figs. S5 and S6).

To quantify temporal changes in correlations, neuron pairs were grouped according to their correlation values on day $t$, and stability was assessed between sessions as the Spearman rank correlation of values on days $t$ and $t+1$. This measure captures the persistence of pairwise relationships over time.

Predictive Effect of Signal Correlations on Future Noise Correlations and Vice Versa (Fig. 3B, C, E, and F and SI Appendix, Fig. S3).

To test whether signal correlations predict future noise correlations, neuron pairs were grouped by noise correlation on day $t$ and subdivided by their signal correlation on the same day. Mean noise correlations were then compared between days $t$ and $t+1$ across these subgroups. Analogously, the predictive effect of NC on future SC was assessed by grouping neuron pairs by SC on day $t$ and subdividing by NC on the same day.

Prediction Index (Figs. 3G–I and Fig. 6B–I and SI Appendix, S5, S6, S7, and S9).

The prediction index quantified whether SC at day $t$ predicts the ordering of NC at day $t+1$. For each NC bin, neuron pairs were grouped into $K$ bins based on their SC at day $t$. For each SC bin $k$, we computed the mean across pairwise NC $Y_i$ as

$$\begin{array}{c}\hfill C_k^{(t)}=\frac{1}{N_k}{\displaystyle \sum _{i\in {\mathcal{I}}_k}}{Y}_i^{(t)},C_k^{(t+1)}=\frac{1}{N_k}{\displaystyle \sum _{i\in {\mathcal{I}}_k}}{Y}_i^{(t+1)},\end{array}$$ [2]

where ${\mathcal{I}}_k$ is the set of pairs in bin $k$. The NC profile’s spread at each time point,

$$\begin{array}{c}\hfill S^{(t)}=C_K^{(t)}-C_1^{(t)},S^{(t+1)}=C_K^{(t+1)}-C_1^{(t+1)},\end{array}$$ [3]

measures how strongly NC varies with SC. The prediction index is the change in spread, ${\chi }_{\text{pred}}=S^{(t+1)}-S^{(t)}$, with positive values indicating that SC at day $t$ predicts a more ordered NC structure at day $t+1$. Then, we computed the prediction index with switched contingencies (NC $\to$ SC).

Conditional Correlation Analysis (Figs. 4 and Fig. 5 and SI Appendix, S4, S8, and S9).

Neuron pairs were grouped according to their SC (or, in a complementary analysis, by NC) on day $t$, and for each group, the Spearman rank correlation of the other measure (NC or SC, respectively) between consecutive days was computed. The slopes from linear fits quantified the stabilizing influence of SC on NC (and vice versa).

Shuffled Control Procedure (Figs. 3 H and I and Fig. 4B, C, E, and F).

To generate shuffled controls, we randomized the assignment of neuron pairs to SC or NC bins depending on the contingency tested. While keeping the total number of neuron pairs per bin fixed, bin labels were randomly reassigned across all neuron pairs, preserving overall correlation distributions but disrupting the structured pairing between signal and noise correlations.

Network Model (Fig. 6 and SI Appendix, Figs. S6–S9).

We employed a linear recurrent network model given by Eq. 1. Network size was set to $N=250$. 34 different stimuli (drawn from a normal distribution) were presented each for 20 trials, analogously to the experiment. The symmetric connectivity $M$ was normalized to maximal eigenvalue ${\lambda }_{max}=0.85$. All solutions were computed at steady state.

Network Initialization for Plasticity Analyses (Fig. 6 and SI Appendix, Fig. S6).

We simulated three different plasticity scenarios, described in the following: I) stochastic drift, II) Hebbian-like plasticity, and III) stochastic drift and Hebbian-like plasticity. Initializations were chosen to keep the statistics of the weight matrix similar across connectivity changes.

Stochastic Drift in the Model (Fig. 6C–E and G–I and SI Appendix, Figs. S6–S9).

To model ongoing synaptic and input variability, stochastic changes were implemented as an Ornstein–Uhlenbeck process (41):

$$\begin{array}{c}\hfill \text{d}X=\frac{{\omega }^2}{2{\sigma }^2}\left(\mu -X\right)\text{d}t+\omega \text{d}W,\end{array}$$ [4]

where $\mu$ and ${\sigma }^2$ are the mean and the variance of the steady state distribution, $\omega$ is a free parameter that controls the strength of drift and $W$ is a standard Wiener process (52). We applied this process either to input vectors $s_i$, to recurrent weights $M_{\mathit{ij}}$ (symmetric), or to both simultaneously.

Hebbian-Like Plasticity in the Model (Fig. 6F–I and SI Appendix, Figs. S6–S9).

We implemented Hebbian-like plasticity in recurrent connections such that coactive neurons strengthened their mutual weights according to

$$\begin{array}{c}\hfill M_{\text{new}}=(1-\gamma )M_{\text{old}}+\gamma \frac{\alpha }{N_{\text{stim}}}{\displaystyle \sum _{k=1}^{N_{\text{stim}}}}{s}_k{s}_k^T,\end{array}$$ [5]

where $M_{\text{new}}$ is the new connectivity matrix, $M_{\text{old}}$ is the previous connectivity matrix, $s_k$ are the presented stimuli, and $\gamma$ is the learning rate. The constant $\alpha$ was chosen such that the SD of $\frac{\alpha }{N_{\text{stim}}}\sum _{k+1}^{N_{\text{stim}}}{s}_k{s}_k^T$ was equal to that of $M_{\text{old}}$. This update maintains the symmetry of $M$.

Models with Hebbian-Like Plasticity and Stochastic Drift (Fig. 6 H and I and SI Appendix, Figs. S6–S9).

In models with both Hebbian-like plasticity and stochastic drift in the recurrent connections, the network weights were adapted by adding a matrix $\text{d}\mathrm{\Omega }$ to the right-hand side of Eq. 5:

$$\begin{array}{c}\hfill M_{\text{new}}=(1-\gamma )M_{\text{old}}+\gamma \frac{\alpha }{N_{\text{stim}}}{\displaystyle \sum _{k=1}^{N_{\text{stim}}}}{s}_k{s}_k^T+\text{d}\mathrm{\Omega }.\end{array}$$ [6]

The elements of $\text{d}\mathrm{\Omega }$ represent changes due to an OU-process and were computed at each update step as

$$\begin{array}{c}\hfill \text{d}{\mathrm{\Omega }}_{\mathit{ij}}=\frac{{\omega }^2}{2{\sigma }^2}\left(\mu -M_{\mathit{ij}}^{\text{old}}\right)\text{d}t+\omega \text{d}W.\end{array}$$ [7]

Model Statistics (Fig. 6 and SI Appendix, Figs. S6–S9).

For each parameter combination, we computed the prediction index and conditional stability metrics analogously to the experimental data. This was done for 100 update steps (each modeling the transition from one imaging day to the next) and repeated for 10 initializations of $M$. Bootstrapping was used to estimate variability, and comparisons to shuffled controls assessed statistical significance.

Comparison Between Prediction Indices (Fig. 3 H and I and SI Appendix, Fig. S5).

Prediction indices were tested for significance using a two-sided Wilcoxon signed rank test. P-values were Bonferroni corrected. The star denotes significance *$P<0.05$.

Comparison Between Parameter fits (Figs. 4B, C, E, and F and Fig. 5A–D).

To test for significant differences, a distribution of parameter values was generated using a bootstrapping procedure. $N=100$ surrogate datasets were generated by randomly sampling neuron pairs with replacement from the original dataset. The conditional correlation analysis was performed on each of the surrogate datasets resulting in a distribution of slopes. Significance was tested using a two-sided Wilcoxon–Mann–Whitney test. P-values were Bonferroni corrected. Star denotes significance *$P<0.05$.

Ethical Statement.

All animal experiments were performed in accordance with the Austrian laboratory animal law guidelines for animal research and had been approved by the Viennese Magistratsabteilung 58 (Approval M58/00236/2010/6).

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

Calcium imaging data time series and analysis code have been deposited in https://doi.org/10.12751/g-node.vnj8ip (53). Previously published data were used for this work (https://doi.org/10.12751/g-node.trwj8c) (38).