Learning an efficient place cell map from grid cells using non-negative sparse coding

Experimental studies of grid cells in the Medial Entorhinal Cortex (MEC) have shown that they are selective to an array of spatial locations in the environment that form a hexagonal grid. However, in a small environment, place cells in the hippocampus are only selective to a single-location of the environment while granule cells in the dentate gyrus of the hippocampus have multiple discrete firing locations, but lack spatial periodicity. Given the anatomical connection from MEC to the hippocampus, previous feedforward models of grid-to-place have been proposed. Here, we propose a unified learning model that can describe the spatial tuning properties of both hippocampal place cells and dentate gyrus granule cells based on non-negative sparse coding. Sparse coding plays an important role in many cortical areas and is proposed here to have a key role in the navigational system of the brain in the hippocampus. Our results show that the hexagonal patterns of grid cells with various orientations, grid spacings and phases are necessary for model cells to learn a single spatial field that efficiently tile the entire spatial environment. However, if there is a lack of diversity in any grid parameters or a lack of cells in the network, this will lead to the emergence of place cells that have multiple firing locations. More surprisingly, the model shows that place cells can also emerge even when non-negative sparse coding is used with weakly-tuned MEC cells, instead of MEC grid cells, as the input to place cells. This work suggests that sparse coding may be one of the underlying organizing principles for the navigational system of the brain.


Introduction
1 The brain can perform extremely complex spatial navigation tasks, but how the brain 2 does this remains unclear. Since the Nobel-prize-winning discovery of place cells in the 3 hippocampus O'Keefe, 1976;O'Keefe and Dostrovsky, 1971) and 4 grid cells in the Medial Entorhinal Cortex (MEC) (Hafting et al., 2005;Rowland et al., 5 2016), brain regions involved in spatial awareness and navigation have attracted much 6 attention from both experimental and computational neuroscientists. Experimental rat studies show that hippocampal place cells have a single specific 8 location in the environment at which they have an elevated firing rate (O'Keefe and 9 Dostrovsky, 1971) and neighboring cells have firing fields at different locations of the 10 environment, such that the local cell population in the hippocampus can represent the 11 whole spatial environment (O'Keefe, 1976). In contrast, granule cells in the dentate 12 1/25 gyrus of the hippocampal formation have multiple discrete firing locations without 13 spatial periodicity (Jung and McNaughton, 1993;Leutgeb et al., 2007). However , Park 14 et al. (2011) also showed that rat place cells can have multiple firing locations in large 15 environments. 16 MEC grid cells are also spatially tuned to the locations of the environment. However, 17 unlike hippocampal place cells, firing fields of grid cells form a hexagonal grid that 18 evenly tile the entire environment (Hafting et al., 2005). The hexagonal grid of each 19 grid cell is characterised by spacing (distance between fields on the grid), orientation 20 (the degree of rotation relative to an external reference), and phase (offset relative to an 21 external reference). The spacing of the grid increases step-wise monotonically along the 22 dorsal-ventral axis (Hafting et al., 2005). Moreover, the progression in grid spacing 23 along the dorsal-ventral axis is geometric, with ratio around 1.42, such that grid cells 24 are organised into discrete modules according to their spacing (Stensola et al., 2012). 25 Additionally, grid cells in each module also have similar orientation but random phases 26 (Stensola et al., 2012). 27 Experimental evidence indicates that MEC grid cells are the main projecting 28 neurons to the dentate gyrus and CA3 of the hippocampus (Leutgeb et al., 2007;29 Steward and Scoville, 1976;Tamamaki and Nojyo, 1993;Zhang et al., 2013). 30 Consequently a variety of models have been proposed to explain the emergence of the 31 firing fields of hippocampal place cells based on the feedforward connection from MEC 32 grid cells, from mathematical models that have no learning (de Almeida et al., 2009;33 Solstad et al., 2006) to models with plasticity (Franzius et al., 2007a,b;Rolls et al., 34 2006; Savelli and Knierim, 2010). 35 For the learning models of grid-to-place formation, Rolls et al. (2006) used a 36 competitive learning procedure to learn place cells from grid cell input. However, only 37 approximately 10% of model cells were found to have a single-location place field. 38 Furthermore, the competition in the model was introduced by manually setting the 39 population activation to a small specified value that indicates a sparse network. 40 Similarly, Franzius et al. (2007b) applied independent component analysis (ICA) 41 (Hyvarinen, 1999) to maximising the sparseness of the model place cells. However, the 42 examples of model place cells in these studies are mostly located at the border of the 43 environment ( Figure 1G in Franzius et al. (2007b) and Figure 3C in Franzius et al. 44 (2007a)). Additionally, in their model the connection strength between grid and place 45 cells can be positive or negative and the place cell responses were manually shifted by 46 the addition of a constant term to ensure that they were non-negative, which puts into 47 question the biological realization of the model. Furthermore, previous models do not 48 investigate how well the learned place map represents the spatial environment. 49 Sparse coding, proposed by Olshausen and Field (1996), provides a compelling 50 explanation of many experimental findings of brain network structures. One particular 51 variant of sparse coding, non-negative sparse coding (Hoyer, 2003), has recently been 52 shown to account for a wide range of neuronal responses in areas including the retina, 53 primary visual cortex, inferotemporal cortex, auditory cortex, olfactory cortex and 54 retrosplenial cortex (see Beyeler et al. (2019) for a review). However, whether sparse 55 coding can account for the formation of hippocampal place cells has not previously been 56 investigated in detail.

57
Here we applied sparse coding with non-negative constraints, where neuronal 58 responses and connection weights are restricted to be non-negative, to building a 59 learning model of place cells using grid cell responses as the input. Our results show 60 that single-location place fields can be learnt that tile the entire environment, given a 61 sufficient diversity in grid spacings, orientations and phases of the input grid cells.

62
However, if there is a lack of diversity in any of these grid parameters, the learning of 63 the place cells is impeded; instead, the learning results in more place cells with multiple 64 2/25 firing locations. Furthermore, a lower number of grid cell inputs results in learning 65 multiple place cell firing locations. The competition generated by the principle of sparse 66 coding in the model naturally provides a global inhibition such that the place cells 67 display discrete firing fields, suggesting that the proposed model can be implemented by 68 biologically based neural mechanisms and circuits. Moreover, the model can still learn 69 place cells even when the inputs to the place cells are replaced by the responses of 70 weakly-tuned MEC cells. This suggests a plausible explanation of why place cells emerge 71 earlier than grid cells during development (Langston et al., 2010;Wills et al., 2010). Sparse coding was originally proposed by Olshausen and Field (1996) to demonstrate 75 that simple cells in the primary visual cortex represent their sensory input using an 76 efficient neuronal representation, namely that their firing rates in response to natural 77 images tend to be sparse (rarely attain large values) and statistically independent. In 78 addition, sparse coding finds a reconstruction of the sensory input through a linear 79 representation of features with minimal error, which can be understood as minimizing 80 the following cost function where the matrix I is the input, columns of A are basis vectors (universal features) from 82 which any input can be constructed from a weighted sum, the vector s represents the 83 neural responses and each element, s i , is the coefficient for the corresponding basis 84 vector, the function Q(·) is a function that penalises high activity of model units, and β 85 is a sparsity constant that scales the penalty function Field, 1996, 1997). 86 The term As in Eq.
(1) represents the model reconstruction of the input, so this cost 87 function represents the sum of squared reconstruction error and response penalty.
88 Therefore, the model finds a sparse representation for the input by solving this 89 minimization problem. By taking the partial derivatives of Eq. (1) in terms of the 90 elements of A and s, and then applying gradient descent, the dynamic equations and 91 the learning rule are given by where · is the average operation, Q (·) is the derivative of Q(·), and the dot notation 93 represents differentiation with regard to time.

94
Non-negative sparse coding is simply sparse coding with non-negative 95 constraints, i.e., the connection weights A and model responses s are restricted to 96 non-negative values in the cost function Eq. (1). Note that, when β in Eq. (1) is set to 97 zero, the cost function of non-negative sparse coding reduces to the cost function of 98 non-negative matrix factorization (Lee and Seung, 1999 The hexagonal firing fields of grid cells are represented in this study by the sum of three 107 sinusoidal gratings (de Almeida et al., 2009;Kropff and Treves, 2008;Solstad et al., 108 2006), as described by where G( r) is the grid cell response at the spatial location r = (x, y), λ is the grid 110 spacing, θ is the grid orientation, r 0 = (x 0 , y 0 ) represents the phase offset, and 111 u j = (cos(2πj/3 + θ)), sin(2πj/3 + θ)) is the unit vector with direction 2πj/3 + θ. G(·) 112 described in Eq.
(3) is normalised to have a maximum value of 1 and minimum of 0.

113
Because of the periodicity of the hexagonal pattern, the grid orientation, θ, lies in the 114 interval of [0, π/3), and the phase offsets in both x and y axes are smaller than the grid 115 spacing, i.e., 0 ≤ x 0 , y 0 < λ.

116
Since grid cells have different spacings, orientations and phases, Eq.
(3) is used to 117 generate diverse grid fields. The value of the grid spacing, λ, ranges in value from 28 cm 118 (Hafting et al., 2005;Solstad et al., 2006) and increases by a geometric ratio 1.42 that is 119 consistent with experimental results (Stensola et al., 2012), and the optimal grid scale is 120 derived by a mathematical study (Wei et al., 2015). x 0 = 0 and λ/2. The resulting total number grid cells, denoted as N g , will be the 130 product of numbers of spacings, orientations and phases: Some examples of grid fields described by Eq.
(3) are shown in Figure 1. These grid 132 fields have diverse grid spacings, orientations and phases. Since the environment is represented by a 32 × 32 grid, a 1024 × 1 vector, denoted by 134 g, can be used to represent the firing field of the grid cell over the entire environment. 135 For a given position r in the environment, the response of a grid cell is simply g T r.

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The number of different grid parameters (N λ , N θ , N x and N y ) of grid cells defined 137 above are assigned to different values to investigate the effect of the diversity of grid 138 cells on the formation of place cells. Next we define grid cells with parameters that 139 better capture the biologically observed variability, which will be used to investigate the 140 robustness of the model.  Because grid phases, (x 0 , y 0 ), are random in each module (Stensola et al., 2012), grid 151 phase is randomly sampled from a uniform distribution. Stensola et al. (2012) showed 152 that 87% of grid cells belong to the two modules with small spacings. Therefore, when 153 using realistic grid fields in the study, we have 43.5%, 43.5%, 6.5% and 6.5% of grid 154 cells in the modules with mean spacings 38.8 cm, 48.4 cm, 65 cm and 98.4 cm, 155 respectively (unless otherwise noted).

156
The firing field of the grid cells is taken to be the sum of the spatial firing pattern at 157 every vertex on the hexagonal pattern. The spatial firing pattern with vertex (x v , y v ) is 158 described by a function with the following form (Neher et al., 2017) 159 where γ v is the amplitude, σ determines the radius of the firing field, and the response 160 will be 1/5γ v at a distance σ away from the center. The amplitude at every vertex of 161 the hexagonal pattern, γ v , is chosen from a normal distribution with mean 1 and 162 standard deviation 0.1, and σ is determined by the grid spacing, λ, with σ = 0.32λ 163 (Neher et al., 2017). The grid field is then the sum of firing fields (described by Eq. 5) 164 at all vertices of the hexagonal pattern. The locations of vertices of the hexagonal 165 pattern are determined by grid spacing, λ, grid orientation, θ, and grid phase, (x 0 , y 0 ). 166

Structure of the model 167
In this study a two-layer network is proposed to model the activities of grid cells (first 168 layer) and place cells (second layer), respectively. Given a spatial location in the 169 environment, grid cells respond according to their firing fields. Grid cell responses then 170 feed into place cells and the grid-place network implements a sparse coding model with 171 non-negative constraints. The model structure is shown in Figure 2.

172
Denote G as a 1024 × N g matrix that represents the firing fields for N g grid cells in 173 the network; i.e. each column of G, g i (i = 1, 2, ..., N g ) is a 1024 × 1 vector that 174 represents the firing field of grid cell i. For a spatial location r in the environment, grid 175 cell responses (firing rates), s g , are given by s g = G T r. Place cell responses (firing 176 rates), s p , are computed by a sparse coding model for the grid-place network with 177 non-negative connection A. Assume there are N p place cells in the network. Then A is 178 a N g × N p matrix and s p is a N p × 1 vector. Denote u p as a N p × 1 vector that 179 Figure 2. Graphical representation of the model. Red arrows represent nonnegative connections. Notation is defined in the main text.
represents membrane potentials of place cells. The model dynamics is given by where τ is the time constant for place cells, β is the threshold of the rectifying function 181 of firing rates, and W can be interpreted as the matrix of recurrent connections 182 between place cells. In this paper, we take identity matrix. The dynamics of place cells described in Eq. (6)  taken to be non-negative in this study.

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The code to run the model is available online (https://github.com/lianyunke/

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Learning-Place-Cells-from-Grid-Cells-Using-Nonnegative-Sparse-Coding). The learning rule for updating the connection strength matrix A is similar to that in 191 previous studies of sparse coding (Olshausen and Field, 1997;Zhu and Rozell, 2013), as 192 given by where η is the learning rate. Elements of A are kept non-negative during training, i.e., 194 the element will be set to 0 if it becomes negative after applying the learning rule 195 described in Eq. (7). Then each column of A is normalised to unit length, similar to 196 previous studies (Lian et al., 2019;Olshausen and Field, 1997;Rolls et al., 2006;Zhu 197 and Rozell, 2013).

198
The model dynamics and learning rule described in Eqs. (6) and (7)  Since the environment used in this study is 1m×1m, the maximal grid spacing is taken 205 to be smaller than 1m, i.e., 1 ≤ N λ ≤ 4. All possible grid spacings are 28 cm, 39.76 cm, 206 56.46 cm and 80.17 cm. For grid orientation, we have 1 ≤ N θ ≤ 7. For grid phase, we 207 have the same number of phases in each direction and the maximal number is 5, i.e.

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There are 100 model cells at the second layer in our simulations, i.e., N p = 100. The 210 dynamical system described by Eq. (6) is implemented by the first-order Euler method, 211 where the membrane time constant is τ = 10ms, consistent with the physiological value 212 (Dayan and Abbott, 2001), the threshold is β = 0.3, and there are 200 integration time 213 steps with a time step of 0.8ms which we found to provide numerically stable solutions. 214 We use 20, 000 epochs in our training. In each epoch, a random location, r, is presented 215 to the grid cells and the model responses are computed using Eq. (6) and the matrix of 216 connection strengths, A, is updated by Eq. (7). The learning rate, η, is chosen to be 217 0.03. The parameters above were chosen to ensure a stable solution in a reasonable time 218 scale, but the results were found to be robust to moderate changes of these parameters. 219 2.6 Recovering the firing fields of model cells 220 After training, we use the method of reverse correlation to recover the firing fields, 221 denoted as F, of model cells. We present K uniformly sampled random locations, 222 r 1 , · · · , r K , to the model, compute according to Eq. (6) the neural responses of a model 223 cell, s 1 , · · · , s K , and then compute the firing field, F, of this model cell by K = 10 5 is used in this paper. The recovered firing field, F (recovered by Eq 8), is fitted by a function Q(x, y) of the 227 form where γ is the amplitude, σ is the breadth of the firing field, and (x c , y c ) represents the 229 center of the 2D Gaussian function. The built-in MATLAB (version R2020a) function, 230 lsqcurvefit, is used to fit these parameters. The fitting error is defined as the square of 231 the ratio between the fitting residual and firing field. For place cells with a single-location firing field, the field center (x c , y c ) fitted by Eq. (9) 241 indicates the spatial location that the place cell responds to. We measured how well 242 these place cells represent the entire environment using two measures.

243
The first measure is distance to place field, d PF , which indicates the Euclidean 244 distance between each spatial location (p x , p y ) in the environment and the nearest place 245 field, described as If the distance to a place field is large for a location, it means that there are no place 247 fields near this location. Therefore, the distribution of this measure can tell us how well 248 7/25 place fields tile the entire spatial environment. When all spatial locations have small 249 values of this distance to place field, d PF , the entire environment is tiled by the place 250 cells.

251
The second measure is the nearest distance, d ND . For all the centers of place cells 252 with single-location firing field, we define nearest distance as the maximal Euclidean 253 distance of 2 nearest centers of each center (x j , y j ), described as  The distance to place field, d PF , together with nearest distance, d ND , provides 261 quantitative measures of how well the place cells code for the spatial environment.

262
Small values of both measures indicate that place cells can tile the entire environment 263 fairly evenly. For example, if 100 place cells are organised on a 10 × 10 grid that evenly 264 tile the 1m ×1m environment, the nearest distance will be 100/(10 − 1) ≈ 11.11 cm for 265 each place cell and the distance to place field for every location is smaller than 266 11.11/2 ≈ 5.56 cm.  represented by a 32 × 32 pixel-like image, Figure 3B, which shows that the centers of 283 the 100 place cells tile the entire environment without any overlap. In addition, the box 284 plot in Figure 3C shows that any location within the space is within a distance of no 285 more than 8.2 cm from the nearest place fields. The histogram of nearest distance of all 286 100 place cells is displayed in Figure 3D, which shows that the distribution is centered 287 around a mean value of 10.70 cm and standard deviation 0.75 cm. Given that the The connectivity profile between 600 grid cells and 100 place cells is plotted in 295 Figure 4A, which shows that each place cell selects a group of particular grid cells with 296 different weights. As a result, the overall feedforward connection from the spatial 297 environment to the place cells, namely the matrix product GA, has the spatial 298 structure plotted in Figure 4B, which shows that each place cell is selective to one 299 spatial location similar to the recovered firing fields ( Figure 3A). However, GA has 300 strong average offsets, which can be seen from the grey background in Figure 4B. The 301 model of place cells proposed by Solstad et al. (2006) has an inhibition term to balance 302 the excitation so that the place fields are responsive to a single location. As for the 303 model of place cells proposed by Franzius et al. (2007b), an offset constant is added and 304 signs of model units are adjusted in order to achieve single location place fields.

305
Nevertheless, comparing Figure 3A and Figure 4B, we can conclude that the network 306 implemented by sparse coding naturally introduce the competition to inhibit place cells 307 such that they have firing fields similar to those found in experiments. As stated earlier 308 in Materials and Methods, the sparse coding model used in this paper can be    Figure 3B and C, the lack of diversity in grid orientation or grid phase 328 will also cause the model to learn more cells with multiple firing locations ( Figure 5B   Recall that the principle of sparse coding finds a linear representation of the input, 335 namely the grid cell responses. Our results suggest that grid cells with less diversity in 336 grid parameters are not sufficient to well represent the whole environment, so that the 337 system gives an ambiguous representation of the spatial location. Therefore, the diverse 338 grid cells found in the MEC are crucial to the emergence of hippocampal place cells. MEC to the dentate gyrus, so the lack of diversity in afferent grid cells may be one 341 possible factor explaining how cells with multiple firing locations emerge in the dentate 342 gyrus.  with small values of the nearest distance, Eq.(11). Furthermore, as N p increases, the 366 mean nearest distance and field breadth decreases (Figure 7), indicating that the spatial 367 resolution of the neural representation by place cells improves. Figure 7. N p vs. mean radius and N p vs. mean nearest distance. As N p increases, the mean radius and mean nearest distance decrease.  Figure 3). (B) The lack of diversity in grid spacing, orientation or phase (left, middle and right plot) leads to cells with multiple firing locations (similar to Figure 5). (C) Fewer model cells leads to cells with multiple firing locations (similar to Figure 6). (D) Spatial resolution of the neural representation increases as N p increases (similar to Figure 7).

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qualitatively consistent with results shown in Figure 3C (mean 10.70 cm and standard 378 deviation 0.75 cm). Therefore, the learned place cells evenly tile the entire environment. 379 Figure 8B shows that realistic grid fields with less diversity will learn model cells  Similar to Figure 7, the neural representation of the spatial environment has better 396 resolution (smaller radius and smaller nearest distance) as N p increases, as seen from 397 Figure 8D. As discussed in a previous study (Neher et al., 2017), most existing model of place cells 400 cannot produce large place fields, such as CA3 place cells with size around 1225 cm 2 .

401
The model proposed here can generate large place fields by simply having grid cells with 402 large grid spacings as the input to model cells.

403
In this part of the study, only grid cells with grid spacings in the fourth module are 404 used; i.e., the grid spacing is sampled from the normal distribution with mean 98.4 cm 405 and standard deviation 8 cm, grid orientation is sampled from the distribution with  Figure 9A shows 411 that large place fields can emerge after learning. Figure 9C shows that these place cells 412 have radii from 18.71 cm to 21.22 cm (mean 19.68 cm and standard deviation 0.75 cm). 413 Therefore, the size of place fields range from 1099.76 cm 2 to 1414.62 cm 2 . Figure 9B,D 414 and E show that these 18 place cells with large size cover the entire environment rather 415 evenly.

416
Above all, the model can learn place cells if the afferent grid cells have large grid 417 spacings, consistent with experimental evidence that the sizes of grid cells and place 418 cells increase along the dorsal-ventral axis (Fyhn et al., 2007;Kjelstrup et al., 2008) and 419 with topographic entorhinal-hippocampal projections along the dorsal-ventral axis 420 (Dolorfo and Amaral, 1998). Recent experimental evidence shows that the emergence of hippocampal place cells 424 happens earlier in development than grid cells (Langston et al., 2010;Wills et al., 2010). 425 Here we show that even weakly-tuned MEC cells can provide sufficient spatial  Weakly-tuned cells are observed to be abundant in the MEC (Zhang et al., 2013).

431
The weakly-tuned field is generated in the simulation by first assigning a random 432 activation, sampled from a uniform distribution between 0 and 1, to each location, then 433 smoothing the map with a Gaussian kernel with standard deviation 6 cm, and 434 normalizing the map such that the values are between 0 and 1 (Neher et al., 2017).  Figure 10A.

437
The fields of weakly-tuned MEC cells are very different from the periodic pattern of 438 grid cells. Surprisingly, they can nevertheless provide sufficient spatial information such 439 that the model based on sparse coding can decode MEC cell responses and give an 440 accurate representation of the spatial location. Figure 10B shows the firing field of 441 learned place cells. Figure 10C, D and E shows that the centers of place cells evenly tile 442 the entire spatial environment.

443
Compared with Figure 3 and 8A, using weakly-tuned MEC cells instead of grid cells 444 results in learning a hippocampal place-map with less resolution. The mean radius of 445 place fields in Figure 10B (11.45 cm) is larger than Figure 3 and 8A (8.92 cm and 8.69 446 cm, respectively). Furthermore, the nearest distance in Figure 10 Figure 10D suggests that the 450 irregular fields of weakly-tuned MEC cells lead to the less even tiling of place cells.

451
Compared with the model with grid cell input (Figure 3),The average active rate for 452 model cells is 30.68%, much larger than the percentage when grid cells are used (5.59%). 453 Therefore, the place cell map using grid cell input is more efficient and energy-saving.

454
Furthermore, the model is quite robust to noise and an efficient place map can still 455 be learned, even though a relatively strong noise is added to the MEC cell responses in 456 Eq. (6): where n is the Gaussian noise with mean 0 and variance 1, and γ n is the amplitude of 458 the noise. Note that the maximal value of G T r is 1 because G is normalised to have the 459 maximum 1. We find that the model can still learn an efficient map when γ n is 0.3 460 ( Figure S4 Fig). 461 Therefore, the model suggests that place cells emerge earlier than grid cells during 462 16/25 development, in part because the neural system can learn a hippocampal map even 463 when the hexagonal spatial field is not well developed.

464
Sparse coding can learn place cells even though the input cells (MEC cells in this 465 paper) are weakly tuned to the spatial environment. Thus, input cells with stronger 466 spatial selectivity can provide more spatial information so that unique place field can be 467 decoded by sparse coding. Barry and Burgess (2007)  Our results also suggest that these weakly-tuned MEC cells can arise from any form 473 of sensory inputs, such as visual input and auditory input, that encode spatial proposed here can explain a recent experimental study that shows that place cell firings 479 mainly reflect visual inputs (Chen et al., 2019) and another experimental study that 480 suggests homing abilities of mice even in darkness may not need accurate grid cell firing 481 (Chen et al., 2016).

483
In this paper, we applied sparse coding with non-negative constraints to a hierarchical 484 model of grid-to-place cell formation. Our results show that sparse coding can learn an 485 efficient place code that represents the entire environment when grid cells are diverse in 486 grid spacing, orientation and phase. However, lack of diversity in grid cells and fewer 487 model cells leads to the emergence of cells with multiple firing locations, like those cells 488 found in the dentate gyrus. In addition, weakly-tuned Medial Entorhinal Cortex (MEC) 489 cells are sufficient for sparse coding to learn place cells, suggesting that place cells can 490 emerge even when grid cells have not been fully developed. Our work differs from them significantly from previous studies on learning place cells 493 from grid cell input (Franzius et al., 2007b;Neher et al., 2017;Rolls et al., 2006). First, 494 we systematically investigate the influence of the diversity in grid cells upon the 495 formation for place cells. Second, we demonstrate that learned place cells can represent 496 the entire spatial environment well. Third, the same model can produce cells with one 497 firing location, multiple firing locations and large place field size, which can account for 498 the emergence of a range of different observed hippocampal cell types. Fourth, we 499 demonstrate that weakly-tuned MEC cells can also provide sufficient spatial information 500 for the emergence of place cells after learning and the model is very robust to noise.

501
Most importantly, all the results presented in this paper are generated by the same 502 model, namely sparse coding with a non-negative constraint.

503
Though the principle (ICA) used by Franzius et al. (2007a,b) to learn place cells 504 from grid cells is similar to the principle of sparse coding used here, the place cell 505 examples in their paper are mostly near the boundary of the environment. Our model 506 can generate much better results, learn an efficient place map that covers the entire 507 environment, and automatically provide the needed inhibition by the dynamics of sparse 508 coding. One reason is that place cells in a population are not necessarily independent 509 because nearby cells do overlap, so ICA might put a too strong assumption. Also, the 510 17/25 non-negativity introduced in this paper makes the model more similar to the real neural 511 system, which might help the model uncover important biological properties. formation. The active firing rate for model place cells when weakly-tuned MEC cells are 517 used as input (30.68%) is much larger than the rate when grid cells are used (5.59%), 518 suggesting that grid cells are more efficient and thereby reduce the energy required by 519 the neural system. Fiete et al. (2008) proposed that grid cells with different spacings 520 and phases altogether form a residual system that efficiently encode the spatial location. 521 In addition, the triangular lattice of the grid pattern is known to be the solution to the 522 optimal circle packing problem (Thue, 1892) and the geometric scale of grid spacings 523 can represent the spatial environment efficiently (Wei et al., 2015). Our study examines the extent to which sparse coding is as an underlying principle in 526 the navigational system of the brain. However, the current model implies no specific 527 neural circuits for the implementation of the sparse coding, rather it is one of the 528 principles that underlies the formation of the neural circuits. Neurophysiological and 529 anatomical studies suggest that the entorhinal cortex and the hippocampus interact via 530 a loop (Tamamaki, 1997;Tamamaki and Nojyo, 1995;Witter et al., 2014). Therefore, 531 feedforward connections from the entorhinal cortex to the hippocampus, recurrent 532 connections within the hippocampus, and feedback connections from the hippocampus 533 to the entorhinal cortex all play an important role, though their specific contributions 534 to the overall function of the network have not been fully uncovered yet. The proposed 535 model based on sparse coding in this study does not rule out any of the network 536 structures mentioned above, as sparse coding can be implemented in neural circuits 537 either in a feedforward network with recurrent connections (Zylberberg et al., 2011) or a 538 network with feedforward-feedback loops (Lian et al., 2019). The current study does not propose a specific biological neural circuit for implementing 541 sparse coding in the entorhinal-hippocampal region, which is the study of ongoing work. 542 Such a model of these neural circuits would need to take into account the 543 experimentally known networks in this area. Also, other properties of place cells such as 544 phase precession (O'Keefe and Recce, 1993), multiple place fields in large environments 545 (Park et al., 2011) and place map in 3D environments (Grieves et al., 2020) will be 546 investigated in the future. In addition, the model here used prefixed grid cells. We did 547 not attempt here to provide a description for how grid cells emerge, but rather the grid 548 cells are assumed to provide an efficient representation of the environment. It would be 549 interesting to also investigate the role of sparse coding in how grid cells themselves 550 emerge. It is hoped that such future work, which incorporates these aspects of the 551 development process of both grid cells and place cells, will provide further insights into 552 how the navigational system of the brain works. Sparse coding represents just one of a 553 number of possible mechanisms that shape network structures, and much remains to be 554 explored to incorporate other mechanisms, such as those associated with the 555 complexities of metabotropic receptor effects, as discussed in Hasselmo et al. (2020). In this study we examined the role of non-negative sparse coding upon hippocampal 558 place cells that receive input from MEC grid cells. The model showed that both place 559 fields and cells with multiple locations can be learned, depending upon specific network 560 parameters. In addition, the learned place cells give an accurate representation of 561 spatial location. Furthermore, weakly-tuned MEC cells are sufficient to drive 562 hippocampal cells to learn place fields. This study elucidates the role of sparse coding 563 as an important mechanism in the navigational system of the brain. When there is less diversity in grid phase (corresponding to the right plot in Figure 8B). 597 There are 81 place cells out of 100 model cells. The distribution of nearest distance has 598 mean 11.67 cm and standard deviation 1.05 cm. Box plot of distance to place field for all spatial locations in the spatial environment.

607
The black lines at the bottom and top indicate the minimum and maximum, and the 608 bottom edge of the blue box, red line inside the blue box and top edge of the blue box 609 represent 25%, 50% (median) and 75% percentile of the data. In this box plot, there are 610 no outliers. (D) Histogram of nearest distance for place cells.