## Abstract

Recent experiments have revealed that neural population codes in many brain areas continuously change even when animals have fully learned and stably perform their tasks. This representational ‘drift’ naturally leads to questions about its causes, dynamics and functions. Here we explore the hypothesis that neural representations optimize a representational objective with a degenerate solution space, and noisy synaptic updates drive the network to explore this (near-)optimal space causing representational drift. We illustrate this idea and explore its consequences in simple, biologically plausible Hebbian/anti-Hebbian network models of representation learning. We find that the drifting receptive fields of individual neurons can be characterized by a coordinated random walk, with effective diffusion constants depending on various parameters such as learning rate, noise amplitude and input statistics. Despite such drift, the representational similarity of population codes is stable over time. Our model recapitulates experimental observations in the hippocampus and posterior parietal cortex and makes testable predictions that can be probed in future experiments.

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## Data availability

No new experimental data were generated in this study.

Experimental data presented in Fig. 5 are originally described in ref. 10. We used the processed data and MATLAB code, which are available at the Caltech Research Data Repository (https://doi.org/10.22002/d1.1229) to produce these plots.

Experimental data presented in Fig. 6 is extracted from Fig. 2c and d of ref. 9. The data are freely available in ref. 79.

## Code availability

Codes for numerical experiments were written in MATLAB (R2020b). Analysis and figures were made using MATLAB (R2020b) except Fig. 4a,b, which is made by R (version 4.2.0). All codes are available in the GitHub repository https://github.com/Pehlevan-Group/representation-drift.

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## Acknowledgements

This work was supported by the NIH grant 1UF1NS111697-01 (C.P. and S.Q.), the Intel Corporation through Intel Neuromorphic Research Community (C.P.) and a Google Faculty Research Award (C.P.). We thank W. Gonzalez, H. Zhang, A. Harutyunyan and C. Lois from California Institute of Technology for sharing the data on place cell recordings. We thank L. Driscoll, N. Pettit, M. Minderer, S. Chettih and C. Harvey for making the T-maze experimental data available. We are grateful to members of Pehlevan group for helpful discussions, and W. Gonzalez, L. Driscoll and C. Harvey for comments on the manuscript.

## Author information

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### Contributions

This work resulted from the merging of two independent projects at their initial stages: one by S.Q. and C.P. and the other by S.F., D.L., A.M.S and D.B.C. For the current manuscript, S.Q. and C.P. conceived and designed the study with input from S.F., D.L., A.M.S and D.B.C. S.Q. performed the numerical simulations and analytical calculations. S.Q. and C.P. analyzed and interpreted the data with input from S.F., D.L., A.M.S. and D.B.C. S.Q. and C.P. wrote the manuscript with comments from S.F., D.L., A.M.S. and D.B.C.

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*Nature Neuroscience* thanks Adrienne Fairhall, Timothy O’Leary and Alessandro Treves for their contribution to the peer review of this work.

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## Extended data

### Extended Data Fig. 1 Performance of the linear Hebbian/anti-Hebbian network in the PSP task.

(**a**,**b**) The PSP error as quantified by \(\left\| {{{{\mathbf{F}}}}_{t}^{{{\bf{ \top }}}}{{{\mathbf{F}}}}_{t} - {{{\mathbf{U}}}}{{{\mathbf{U}}}}^{{{\bf{ \top }}}}} \right\|_F/\left\| {{{{\mathbf{UU}}}}^{{{\bf{ \top }}}}} \right\|_F\), where **U** is a *n* × *k* matrix whose columns are the top *k* left singular vectors of \({{\mathbf{X}}} \equiv \left[ {{{{\mathbf{x}}}}_1, \cdots ,{{{\mathbf{x}}}}_T} \right]\) and **F**_{t}≡**M**^{−1}_{t}**W**_{t}, drops very quickly during training (**a**) and maintains the low error in the presence of synaptic noise (**b**). (**c**) The relative change of the similarity matrix at time *t* compared to time point 0, corresponding to the point where the network initially learned the task, defined as \(\left\| {{{{\mathbf{Y}}}}_{t}^ \top {{{\mathbf{Y}}}}_{t} - {{{\mathbf{Y}}}}_0^ \top {{{\mathbf{Y}}}}_0} \right\|_{{{\mathrm{F}}}}/\left\| {{{{\mathbf{Y}}}}_0^ \top {{{\mathbf{Y}}}}_0} \right\|_{{{\mathrm{F}}}}\). (**d**) Estimating rotational diffusion constant \(D_{{{\mathrm{\varphi }}}}\) from mean squared angular displacement (MSAD). Gray lines are MSAD estimated based on individual representation trajectory \(\mathbf{y}(t)\). The dashed line is a linear fit between \(\left\langle {\left( {{{\Delta }}\varphi } \right)^2} \right\rangle \equiv \left\langle {\left( {{{{{\varphi }}}}\left( {t + {{\Delta }}t} \right) - {{{{\varphi }}}}\left( t \right)} \right)^2} \right\rangle\) and \(\Delta t\) to estimate the rotational diffusion constant. Inset: illustration of Δφ. Parameters are the same as Fig. 2 in the main text.

### Extended Data Fig. 2 A single output neuron’s RF drift when stimuli lives on a ring.

(**a**) With stimuli living on a ring, a single RF has the shape of a truncated cosine curve, whose centroid drifts on the ring like a random walk. (**b**,**c**) The effective diffusion constant D of the centroid position increases with learning rate η both without explicit synaptic noise (σ = 0) (**b**), and with explicit noise (**c**). Error bars: mean ± SD, n = 40 simulations. Magenta lines correspond to theory Eq. (27) in the main text. (**d**) The single RF with larger amplitude has smaller diffusion constant. The amplitude of RF is varied by changing the value of α. Shading: mean ± SD, n = 40 simulations.

### Extended Data Fig. 3 Distinct contribution of noise from forward synapses and recurrent synapses to representational drift in the 1D place cell model.

To further verify the role of noise in feedforward synapses, we simulated models of representational drift in 1D place cells, and compared the correlation coefficient of population vectors of the principal output neurons in three different noise scenarios: full model with all synaptic noises (blue); noise only in the forward synapses \({{{\mathbf{W}}}}\) (\(\sigma _M = 0\), red); and noise only in recurrent synapses \({{{\mathbf{M}}}}\) (\(\sigma _W = 0\), gray). These models are further explored in main text Fig. 5 and Extended Data Fig. 4. In both the simplified 1D place cell model (**a**) and the more detailed network model with inhibitory neurons (**b**), noise in the forward matrix has much larger influence on the representational drift. For the network with inhibitory neurons, forward noise corresponds to all noises in matrices \({{{\mathbf{M}}}},{{{\mathbf{W}}}}^{EI},{{{\mathbf{W}}}}^{IE}\) are set to 0. Shading: mean ± SD, n = 200 output neurons. Parameters used are in Supplementary Table 1 of SI.

### Extended Data Fig. 4 A Hebbian/anti-Hebbian network model of CA1 with both excitatory and inhibitory neurons exhibits similar representational drift as the network in Fig. 5 of the main text.

(**a**) A Hebbian/anti-Hebbian network with inhibitory neurons derived from a similarity matching objective. The derivation is given SI Section 3. (**b**) Upper: learned place fields tile a 1D linear track when sorted by their centroid positions (left), but continuously change over time (right). Lower: Representational similarity matrix \({{{\mathbf{Y}}}}^ \top {{{\mathbf{Y}}}}\) of position is stable over time. (**c**) Peak amplitude of an example place field during a simulation. (**d**) Due to the drift, the average autocorrelation coefficient of population vectors decays over time. Shading: mean ± SD, n = 200 places, population vectors consist of only excitatory neurons. (**e**) Despite the continuous reconfiguration of place cell ensembles, the fraction of cells with active place fields is stable over time. (**f**) Neurons whose RFs have larger average amplitude is more stable, as characterized by smaller D. (**g**) Probability distribution of centroid drifts of place cells at three different time intervals. (**h**) Same as Fig. 5k in main text. Drifts of RFs show distance-dependent correlations, quantified by the average Pearson correlation coefficient. Shading: mean ± SD, n = 20 repeats. Error bars: mean ± SD, n = 13 animals. Parameters used are in Supplementary Table 1 of SI.

### Extended Data Fig. 5 Drift of 2D place cells in the model.

(**a**) Representational similarity is preserved despite the continuous drift of place cell RFs. Positions on the plane (discretized as 32 × 32 lattice) are represented by an index from 1 to 1024. (**b**) The dynamics of RFs are intermittent. The peak amplitude of an example place field has active and silent bouts. (**c**) The intervals of silent bouts follow approximately an exponential distribution. (**d**) At the population level, there is a constant fraction of active RFs over time. (**e**) Dependence of the effective diffusion constant on the total number output neurons. Error bars: mean ± SD, n = 40 simulations. (**f**,**g**) Place cells that have stronger place fields tend to be active more often (**f**) and also more stable as indicated by smaller diffusion constant (**g**). Parameters used are in Supplementary Table 1 of SI.

### Extended Data Fig. 6 Representational drift in a modified 1D place cell model with alternating learning and forgetting periods.

We introduced a forgetting time scale (1/*η*_{forget}) to our learning rules. The model is described in detail in SI Section 4. (**a**) 100 synaptic updates (shaded region) are sequentially followed by a forgetting period with 500 synaptic updates. Including a slower forgetting time scale significantly enhances the stability of learned representation as quantified by the similarity matrix alignment (RSA), defined in equation (41) of SI (upper). The representational similarity matrices \({{{\mathbf{Y}}}}^ \top {{{\mathbf{Y}}}}\) after the last forgetting period for three different forgetting time scales (lower). (**b**) Place fields of 3 exemplar output neurons in the presence of input and synaptic noise. Time starts from when the system has fully learned the representation. (**c**) Even with slow forgetting time scale, the representation still drifts during ‘experiment’ sessions as shown by the decay of coefficients of population vectors across learning sessions (shaded regions in (**a**)). Parameters are listed in Supplementary Table 1 of SI. Shading: mean ± SD, n = 200 output neurons. In (**a**) and (**b**), \(\eta _{forget} = 10^{ - 3}\).

### Extended Data Fig. 7 A Hebbian/anti-Hebbian network model of the PPC with both excitatory and inhibitory neurons exhibits similar representational drift as the network in Fig. 6 of main text.

(**a**) Population activity of excitatory neurons for the left-turn and right-turn task before (upper) and after (lower) sorting based on the centroids of their RFs. Only neurons that have active RFs at the given time point are shown. (**b**) Population activity drifts but representational similarity is stable over time. Activity of excitatory neurons that are active (either tuned to left turn or right turn) in the sorted time (upper and middle). Representational similarity matrix is stable for both left-turn and right-turn task (lower panels). (**c**,**d**) Comparison of drift statistics between model and experiment, corresponding to panels **d**–**f** of Fig. 6 in the main text. Error bars: mean ± SD, n = 5, 5, 4 mice for ∆ = 1, 10, 20 days.

### Extended Data Fig. 8 Degeneracy of the learning objective function and representational drift.

We compare the long-term behavior of learned representations in three different networks. (**a**) Upper: the Hebbian/anti-Hebbian network for PSP. Lower: the evolution of the three components of a representation \({{{\boldsymbol{y}}}}_t\). (**b**) Upper: The network differs from the Hebbian/anti-Hebbian network only in the recurrent matrix **M** which breaks the rotational symmetry of the PSP solution. The learning rule is the same. Lower: the learned representation is stabilized and only fluctuates around its equilibrium. (**c**) A single feedforward network that perform online principal component analysis with Sanger’s rule^{80}. This network has only feedforward input matrix W and the learning rule is nonlocal. Lower: learned representation is relatively stable in the presence of noise. Parameters are the same as in the Fig. 2 of main text except that η = 0.01.

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Qin, S., Farashahi, S., Lipshutz, D. *et al.* Coordinated drift of receptive fields in Hebbian/anti-Hebbian network models during noisy representation learning.
*Nat Neurosci* **26**, 339–349 (2023). https://doi.org/10.1038/s41593-022-01225-z

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DOI: https://doi.org/10.1038/s41593-022-01225-z