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 employ RNN models under different conditions to model MEC cells with both well-understood (e.g., place cells) and poorly-understood (heterogeneous cells) tuning properties. By training the network to minimize the place cell-mediated path integration, the network performed incredibly well in explaining MEC-cell dynamics. By removing neurons in the model, the network's path-integration performance was found to be at least as dependent on the model's heterogeneous neurons as on the stereotypical ones. The authors then finish by introducing a reward-based foraging model, again finding responses that match experimental results very well.