Explaining heterogeneity in medial entorhinal cortex with task-driven neural networks

35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . 1 Introduction . From exploring new areas , planning shortcuts , and returning to remembered locations , the ability to self-localize within an environment subserves a range of navigational behaviors that are essential for survival . The hippocampus ( HPC ) and medial entorhinal cortex ( MEC ) are known to contain cells that encode the position of an animal by displaying firing fields with strikingly regular response patterns , influenced by self-motion [ O ’ Keefe and Dostrovsky , 1971 , Hafting et al. , 2005 , Kropff et al. , 2015 , Solstad et al. , 2008 , Sargolini et al. , 2006 ] . For example , MEC grid cells possess characteristically symmetric and periodic tuning curves , yielding hexagonal arrays of neural activity over physical space . Additionally , border cells that fire maximally near environmental boundaries and head direction cells that fire only when an animal faces a particular direction are among the other MEC cell types with interpretable tuning curves . However , a large fraction of the MEC population have unconventional and heterogeneous tuning to navigational variables and are less obviously well-described in terms of simple , stereotypical tuning patterns [ Hinman et al. , 2016 , Hardcastle et al. , 2017 ] . How can we characterize what these other , more heterogeneous , populations of cells do ? Are they critical to the abilities of the animal in real-world memory tasks ? And if so , how is their role distinct from the more stereotypical grid-like cells ? Here , we take a quantitative computational modeling approach to answering these questions . To start with , we address the question of whether there is a phenomenon to model in the first place , and how one would measure model accuracy quantitatively . To this end , we identify the similarity transform between neural populations in different animals — that is , a mapping which takes neuronal population vectors in a “ source ” animal and maps it to corresponding neuronal population vectors in a “ target ” animal . To the extent that robust similarity transforms can be found that match up MEC population responses across multiple animals measured in multiple experimental conditions , then a reliable pattern of neural response behaviors ( across conditions ) has been identified . This identification strategy is well-defined even when there is no known a priori taxonomy of functional response types . We find here that with the right mapping class ( mostly Ridge-like linear regression ) , the inter-animal consistency of MEC neuronal responses is in absolute terms very high ( > 0.8 ) , and in relative terms just as high as for heterogeneous cells as for more stereotypical grid cells . Using this same similarity transform , we then evaluate the ability of each of multiple computational models to explain response variance of MEC neurons , treating each candidate model as a potential “ source animal ” and measuring how well it maps to each target real animal . We look to the recent literature to identify potential candidate models . Over the past several decades , collaborations between experimental and computational neuroscientists have led to the formulation of dynamicalsystems models of grid cell formation , and helped illustrate possible functional roles for grid cells in supporting path integration-based hippocampal place cells [ Skaggs et al. , 1992 , Zhang , 1996 , Fuhs and Touretzky , 2006 , Burak and Fiete , 2009 ] . These powerful models make a number of nonobvious predictions about MEC neural properties , some of which have been confirmed in subsequent experimental work [ Ocko et al. , 2018 , Campbell et al. , 2018 ] . Despite their success , such models are limited in their explanatory scope , hand-designed to capture the properties of one stereotypical cell-type class ( e.g . grid or border cells ) at a time , or combinations of several cell types via multiple dedicated type-specific modules [ Couey et al. , 2013 , Yoon et al. , 2013 ] . A potential solution to this problem arises out of recent work creating learned neural networks that achieve path integration [ Cueva and Wei , 2018 , Banino et al. , 2018 ] . Intriguingly , these models have been found to contain internal units that resemble grid cells , suggesting that such stereotypical cells embody a computational solution to path integration that naturally arises from satisfying an end-toend functional constraint . Recent work has demonstrated that the underlying mathematical reason for this fact is due to pattern forming dynamics under a nonnegativity constraint [ Sorscher et al. , 2019 , 2020 ] . Intriguingly , in addition to having units that resemble stereotypical grid or border cells , these learned neural networks also naturally possess a wide variety of other less easily described unit types , raising the possibility that these artificial “ heterogeneous ” units might be somehow resemble the actual heterogeneous cells making up the majority of real MEC populations . Motivated by these ideas , we generate a wide variety of candidate model networks by varying architectural structure and end-to-end optimization objectives , each expressing a different hypothesis for MEC circuit structure and function . Architecturally , we formulate variants based on using different types of nonlinearities and different types of local recurrent memory circuits ( e.g . RNNs [ Elman , 1990 ] , UGRNNs [ Collins et al. , 2017 ] , GRUs [ Cho et al. , 2014 ] , and LSTMs [ Hochreiter and Schmidhuber , 1997 ] ) . From a task point of view , we test both simple dimensionality reduction [ Stachenfeld et al. , 2014 , Dordek et al. , 2016 ] as well as place cell mediated vector path-integration [ Banino et al. , 2018 ] and direct position estimation [ Cueva and Wei , 2018 ] . Our core result is that there is substantial variation in the models ’ abilities to explain MEC responses , especially those of the heterogeneous cells , as a function of a model architecture and task . Some models , such as the classic “ Grid Cell ” model based on low-rank decomposition of place cell fields , do a reasonable job explaining grid cell responses but are quite poor at explaining most other neurons . Task-optimized learned models typically do better , especially those optimized for place cell-mediated vector path-integration . The best model – with memory-gated rectified nonlinearities – essentially solves the neurons , capturing nearly 100 % of the noise ceiling of the data . This is of substantial interest , given that the nonlinear components of this model are not directly optimized to match neural data ( just to solve the task ) , and given that a series of strong control models capture much less of the MEC neural variability . This same model also best generalizes to a variety of novel experimental conditions , and has the best match to the empirical data on a grid score distribution metric . With this predictive model in hand , we then begin to address our second core question : what is the functional role of heterogeneous neurons ? We generate several results suggesting that heterogeneous neurons are important for path integration , including cell-type specific virtual knockout experiments , in which we compare performance degradation when deleting grid and/or border cells as compared to heterogeneous units . Overall , we find that models are quite robust to knockouts , and differences between heterogeneous and stereotypical cell knockouts are very small , suggesting that stereotypical cell-types may not be especially more functionally important than units with less easily-characterized response profiles . Building on the above results , we extend models to encompass MEC cell responses as a function of reward as well as spatial position , introducing a simple modeling paradigm that performs rewardmodulated foraging in the context of the path-integration task . We find that this unified task-optimized model matches neural responses across all reward and spatial conditions , and that a reward-response mechanism at an intermediate point on the explore-exploit continuum best explains neural responses . Taken together , our results suggest how specific processes of biological performance optimization may have directly shaped the neural mechanisms in MEC as a whole , and provides a path for enlarging the study of MEC beyond overly-restrictive response stereotypes . 2 Reliability of heterogeneous cell response profiles . What firing patterns of MEC cell populations are common across multiple animals , and thus worthy of computational explanation ? This question is comparatively straightforward for stereotypical MEC cells , because the very presence of these stereotypical features ( e.g . hexagonal grids of a given orientation and spatial frequency ) allows the definition and measurement of observables that arise reliably across trials and animals ( e.g . the grid score distribution ) . But how can this be done generally for populations in which one does not have a prior characterization of what each cell encodes ? We take inspiration from methods that have proven useful in modeling visual , auditory , and motor cortex [ Yamins and DiCarlo , 2016 , Kell et al. , 2018 , Michaels et al. , 2020 ] . Specifically , we aim to identify the narrowest class of similarity transforms needed to map the firing patterns of one animal ’ s MEC population to that of another ( Fig . 1a ) . As with other cortical areas , this transform class likely can not be so strict as to require fixed neuron-to-neuron mappings between MEC cells , since even within the same animal at different times , MEC and HPC populations can undergo remapping that shift cell responses across the population [ Farhoodi et al. , 2020 , Low et al. , 2020 ] . However , the transform class for MEC also can not be so loose as to allow a completely unconstrained linear mapping , since the highly structured response patterns of stereotypical MEC cell types such as grid cells may not be guaranteed to be preserved under arbitrary linear transforms . To identify this transform class , we utilize data collected from electrophysiology in 12 awake behaving mice ( n = 620 cells ) performing open foraging for randomly scattered crushed cereal in a 100cm2 2D arena [ Mallory et al. , 2021 ] . We explore a variety of mapping transform classes between the population rate maps of these neurons ( Fig . 1b ) . As a baseline we evaluate a strict one-to-one mapping transform , in which each target unit is mapped to the single most correlated unit in the source animal . We also evaluate more powerful linear transforms , including Lasso , Ridge , and ElasticNet regression , as well as population-level Partial Least Squares ( PLS ) regression . For all methods with fittable parameters , mapping fit is performed on a random 20 % of spatial position bins , and evaluated on the remaining 80 % of position bins ( see supplement for more details ) . The strict one-to-one mapping yielded low inter-animal consistency among the maps considered , capturing around 50 % of the target neural response variability . Linear regression with strong sparseness priors , such as Lasso ( L1 penalty ) and balanced Lasso-Ridge ( equal L1 and L2 penalty ) regression , also proved to be too strict when evaluated with fixed regularization level α = 1 , yielding hardly any improvement over the one-to-one mapping . However , pure Ridge ( L2 penalty ) regression at this regularization level was highly effective , recovering nearly all target variability for most neurons . Cross-validating the L1 regularization constant on a per-cell basis improved the fits , at the cost of requiring substantially looser regularization than for L2 ( Fig . 1c ) . Under an ElasticNet mapping in which both sparsity penalty and regularization constants were chosen with cross validation , most cells were generally still best explained by an essentially Ridge-like transform with no sparsity penalty ( Fig . 1d ) . However , different target cells required different numbers of source units to achieve effective mapping ( Fig . 1e ) . As expected , cells with more stereotypically grid-like response patterns on average chose a higher sparsity penalty ( slight positive slope in Fig . 1d ) and required fewer source cells to capture ( slight negative slope in Fig . 1e ) as compared to heterogeneous cells , though this effect is weak . Critically , heterogeneous cells did not have lower inter-animal consistency than grid cells ( Fig . 1f ) . These results show that the heterogeneous non-stereotyped cell populations are reasonably similar across animals – at least up to ( mostly Ridge-regularized ) linear transform – establishing that there is a reliable target pattern to study in the first place .

Summary: The authors used linear regression to first demonstrate the robustness of non-grid-like responsive cells in experimental recordings of MEC cells. They then used the same regression to assess the similarity of responses derived from the goal driven training of RNNs to MEC responses. They found that responses derived from RNNS contained these non-grid-like responses and explained neural activity better than a low-rank decomposition of place fields.  These non-grid-like responses were crucial for path integration and likely explained the difference in similarity matching between goal-driven training and low rank decomposition.