Trading Place for Space: Increasing Location Resolution Reduces Contextual Capacity in Hippocampal Codes
Authors: Spencer Rooke
Zhaoze Wang
Ronald W Di Tullio
Vijay Balasubramanian
Venue: NeurIPS 2024 oral (2024)
Keywords: Neuroscience
Neural Coding
Memory
Primary area: neuroscience_and_cognitive_science
TL;DR: We characterize the capacity of the place system to distinguish context, as well as the tradeoff between this ability and the ability to determine location.

Abstract: Many animals learn cognitive maps of their environment - a simultaneous representation of context, experience, and position.  Place cells in the hippocampus, named for their explicit encoding of position, are believed to be a neural substrate of these maps, with place cell "remapping" explaining how this system can represent different contexts. Briefly, place cells alter their firing properties, or "remap", in response to changes in experiential or sensory cues. Substantial sensory changes, produced, e.g., by moving between environments, cause large subpopulations of place cells to change their tuning entirely. While many studies have looked at the physiological basis of remapping, we lack explicit calculations of how the contextual capacity of the place cell system changes as a function of place field firing properties. Here, we propose a geometric approach to understanding population level activity of place cells.  Using known firing field statistics, we investigate how changes to place cell firing properties affect the distances between representations of different environments within firing rate space.  Using this approach, we find that the number of contexts storable by the hippocampus grows exponentially with the number of place cells, and calculate this exponent for environments of different sizes. We identify a fundamental trade-off between high resolution encoding of position and the number of storable contexts. This trade-off is tuned by place cell width, which might explain the change in firing field scale along the dorsal-ventral axis of the hippocampus. We demonstrate that clustering of place cells near likely points of confusion, such as boundaries, increases the contextual capacity of the place system within our framework and conclude by discussing how our geometric approach could be extended to include other cell types and abstract spaces.

Trading Place for Space: Increasing Location Resolution Reduces Contextual Capacity in Hippocampal Codes
Spencer Rooke'
Zhaoze Wang² Ronald W. Di Tullio Vijay Balasubramanian 1.4,5
*Departments of Physics 2 of Computer and Information Science and 3 Neuroscience
University of Pennsylvania; 'Rudolf Peierls Centre for Theoretical Physics, University of Oxford;
and Santa Fe Institute srooke @ www.upenn.......ee.c edu ron. @wn. gmmil..com
*: Equal contribution
zhaoze @ seas, ceeas.upennedii vijay @ wwwsics.upenn.cce
Abstract
Many animals learn cognitive maps of their environment a simultaneous represen- tation context, experience, and position. Place cells in the hippocampus, named for their explicit encoding of position, are believed to be a neural substrate of these maps, with place cell "remapping" explaining how this system can represent different contexts. Briefly, place cells alter their firing properties, or "remap", in response changes in experiential sensory cues. Substantial sensory changes, produced, e.g., by moving between environments, cause large subpopulations of place cells change their tuning entirely. While many studies have looked at the physiological basis of remapping, we lack explicit calculations of how the contex- tual capacity of the place cell system changes as a function of place field firing properties. Here, we propose geometric approach to understanding population level activity of place cells. Using known firing field statisties, we investigate how changes to place cell firing properties affect the distances between representations different environments within firing rate space. Using this approach, we find that the number of contexts storable by the hippocampus grows exponentially with the number place cells, and calculate this exponent for environments of different sizes. We identify fundamental trade-off between high resolution encoding of position and the number storable contexts. This trade-off is tuned by place cell width, which might explain the change in firing field scale along the dorsal- ventral axis of the hippocampus. We demonstrate that clustering of place cells near likely points of confusion, such as boundaries, increases the contextual capacity of the place system within our framework and conclude by discussing how our geometric approach could be extended to include other cell types and abstract spaces.
Introduction
Decades of experiments suggest that the mammalian hippocampus is crucial for the formation of episodic memories and spatial navigation [I] 2 3). Neural recordings of rodents during active navigation led the discovery of place cells by John O' O'keefe 4, named for their spatially localized firing patterns. These place cells were quickly theorized to be the substrate of the cognitive map an animal's simultaneous and abstract representation of context, experience, and position 3 (). Further experiments led to the discovery of remapping ( 7 8) during which place cells alter their firing properties in response to changes sensory and contextual cues. Large contextual changes lead to global remapping. in which population level maps of activity appearing different contexts are nearly orthogonal, independent correlations within an environment 2 10). Many have speculated that
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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Figure 1: Place cell firing fields remap in one (A) and two (C) dimensions. (B) The maps fA. fB of one dimensional contexts correspond to curves neural population activity space, parametrized by position. With constant Gaussian noise, we require that these curves be distance 20 20r(VN + q) apart in order to discriminate contexts. (D) The maps fA. fB of two dimensional contexts correspond to surfaces in activity space, parametrized by position physical space. With activity dependent, Poisson-like noise, firing patterns exponentially localize to characteristic ellipsoids. We require that these thickened surfaces do not intersect in order to discriminate contexts.
the place cell system encodes context via global remapping and that this encoding scheme should be able to store large number of contexts [[], but precise calculations backing these speculations are lacking. Here, analyze the geometry of hippocampal codes to approximate the capacity and properties context encoding by place cells.
We treat place cell population activity as a high dimensional space into which we can embed contexts (Fig. 1 ,B). Since a large fraction of hippocampal neurons are place cells [I3 [4], we expect this activity space to have thousands of dimensions, depending on the species. While the defining features that distinguish context are not fully known 7 [J we define as a choice of
surrounding environment, as is often done in practice in experiments [9] 15 16 17
[8] 19).
Embedding an environment in the firing rate space produces a neural manifold, with position on the manifold corresponding to particular patterns of neural population activity that reflect location in real space. Without noise, the dimensionality of this manifold matches the dimensions of the encoded environment. For example, different linear tracks would be encoded different ID curves embedded in the population activity space (Fig. 1B). In this geometric framework, the distance between two manifolds indicates how differently the neural populations encode each environment 20 (Fig. IB); and if two manifolds do not overlap at any point, the underlying context can always be determined without confusion. In the noiseless case, we expect that it trivial to discriminate context because pairs of very low dimensional manifolds are unlikely to overlap in such a high dimensional space. Real neurons, however, will have noisy responses which "blur" our manifolds, giving them a characteristic width around the noiseless, low dimensional embedding of an environment. As such, our criterion for separability of contexts requires that these thickened manifolds for each context do not overlap extensively in rate space (Fig. 1D). We consider two models of neuronal noise for our rate based neurons: one in which noise is additive and constant, and a second in which noise is additive, but with variance scaling with neural activity. The latter mirrors an underlying Poisson-lit process of spike generation that is often assumed for hippocampal neurons. We investigate the effect of these different noise models on pairwise overlap, and from this extrapolate the probability that multiple contexts are discriminable.
When the number of neurons N is large, these manifolds grow far apart, and are more easily distin- guished. In weak moderate noise regimes, we find that the number of contexts that can be stored by a place-like coding scheme grows exponentially in N, even when we enforce strict requirements that the system can discriminate between contexts at any position within an environment. We further find that place cell width tunes both the ability to decode local position and the typical distance between contextual manifolds. Large place cell widths generate greater overlap between firing fields, which in turn increases discriminability between contexts. Conversely, small place cell widths increase the spatial resolution of encodings, but decrease the separability of contextual manifolds.
This leads to fundamental trade-off between decoding context and local position. We propose that this trade-off accounts for the observed change firing field width across the dorsal-ventral axis in the rodent hippocampus, and predict that selective inhibition along the hippocampus will lead to

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different types of memory impairment for spatial tasks, consistent with existing experimental evidence
24 25 26]. Finally, we find that when fields are uniformly distributed, confusion between contexts is most likely to occur near boundaries. This effect can be compensated for by biasing the density of place field centers towards the boundary, recreating a known feature of rodent place coding [27 28). We predict that the observed place cell clustering near boundaries allows the place system to segregate different contexts with greater efficiency, and the extent of this clustering is additionally function of cell location along the ventral-dorsal axis.
21 [2 23
1.1 Model Description
We consider a population of N place cells, indexed by j. with activity described by a population activity vector F. We treat this vector position within a context (x A). We consider physical environments in one or two dimensions, as this is common experimentally, so that A is either a one two dimensional vector describing location. Each context A is equipped with a place map fA. which defines the tuning of each neuron when the animal is within A. That is, fA sets the mean firing rate of each neuron, (2 A)) fx,((z)). Each map FA can be viewed as an encoding for a particular context that embeds IAA into a certain set of population activity vectors F (Fig. 1A,B). We restrict analysis to rate coding models of population activity, where F" represents population firing rates (rather than spike counts) and has additive noise.
random variable that depends both context (A), and
We assume that place maps are constructed stochastically for each environment, consistently with known properties of place cells. Within an environment, each place cell has aj distinct firing fields, where a, is drawn from a gamma-poisson distribution, following recent experimental observations in rodents [29 30], (Supplemental). For small enviromments (1-2 m), typical place cells have 0-2 firing fields under these statistics, with greater recruitment as environment size increases. We give each firing field gaussian shape, and vary the widths parametrically. The tuning curve of each neuron is then a sum of gaussians:
4j
(A)(EA)= .1H + CJ.A Σ exp
2(w;/2)2 (FA - AA,,)²
For simplicity, the normalization C,.A is chosen so that all neurons have a baseline firing rate of 2HH2, and maximum firing rate of 30H in environments in which they are active.
With additive noise, neural activity is given by (I, A) = (x.) Ey. We consider two noise model. In the first, 5 is gaussian distributed, and noise is not correlated between place cells, so that E~\((O (O, IT)). Note that x² is the variance in noise magnitude and thus has units of H In the second model, we scale the noise variance of each place cell with activity to match underlying Poisson-like -ike statisties associated with spike generation. In this case, Ej (O, of af (x.) Here, & is a dispersion coefficient that sets how noisy the place system is, and has units of Hz. We can write our ( (I A), ødiag A ( ())) respectively. Additive noise can generate negative firing rates, and so we rectify all negative firing rates Hz.
models of place cell activity as F(aTA A) (((((v))) a), IT and F( A)
2 Results
2.1
Place Coding Can Store Exponentially Many Contexts
With the above model in hand, we sought to explore the context coding capacity of the place cell network. To do so, we first defined what it means to discriminate two contexts A and B. In our formulation, fA fA(TA) A) and fa(Ig) define two manifolds the space of neural activity (Fig. 1). Without noise, if these two manifolds do not intersect, then the firing patterns that arise in each environment are unique to that environment. In this case, both position and context can be uniquely determined from population activity, and the contexts A and B are discriminable and fn do not intersect. For the moment, we disregard considerations of computational complexity required for manifold discrimination, such as requiring linear separability as in 31 32
long fA
When there no noise, there a very low probability that the surfaces defined by fa and In intersect any point in our high tigh-dimensional firing rate space, and the intersection criterion is trivial to fulfil.
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The introduction of noise gives the manifolds defined by fA and fB a characteristic width, whose scale and geometry depends on the nature of the noise. In the model where noise variance o² is constant, the characteristic manifold width scales like ovN when N is sufficiently large. This is due to well known characteristic of gaussians in high dimensions, in which normal distributions have the majority of their mass sitting near a thin annulus of radius / RS 34 (Supplemental) That is, the probability density for the radius of vectors pulled from high dimensional spherical gaussians peaks at /N, and falls off exponentially away from this shell as pn(a +g) = Pnlov/N NN For example, when q = 2 this leads an approximate four e-fold decrease in the probability density, so that the majority of the probability mass (> 99% for large N) is within a radius of a ovN- qa. Importantly, this exponential fall off is independent of N, and so the width of the annulus is of order 1 in N. In the rate dependent noise model, we instead get probability mass that is exponentially localized to an ellipsoid with major axes whose lengths depend on firing rates, /NolA(EA)
We would then like a way to determine the effect of both noise models on manifold overlap. and by extension, decrease in context discriminability. We solve this problem geometrically. For the rate independent (gaussian) noise model, the manifold fa (Z acquires width of o VN + q) in every direction. Here, q accounts for the non-zero width of the noise annulus. In any case, our condition that two contexts A and B be distinguishable then becomes requirement that the minimum distance between (Z and fg (TB) in rate space overcomes this width set by noise (Fig. 1C):
min ((A((E))) 20(VN + g) BA
22
The manifold width acquired from the rate dependent (Poisson-like) noise model will vary in each direction of the firing rate space with the neural firing rate. Intuitively, we can think of this widening of the manifold as placing an ellipsoid at each point along our manifold, with principle axes set by the firing rates of each neuron (Fig. 1D). We then check if our thickened manifolds overlap. Unlike the case with constant noise, where it sufficed to check that the distance between points between different manifolds greater than the minimum distance set by noise, we must check that the ellipsoids centered at fa ()) and (T( do not overlap for any pair TA, UB on the two manifolds. For ellipsoids with centers AA (A(A)) and jig = fD(( and covariances ΣA and EB, we can define the set:
E, (s(F TAA) ' AF MA) + -1- (F EB AB) 1
(3)
Here, 8 E [0, 1]. = = 1, this set describes the interior of the ellipsoid centered at fa or nū, respectively, and varying s interpolates between the two. For other values of s, &, is either empty, single point, or the interior of an ellipse. We also note that, for any 8, the intersection of the two ellipsoids is always contained in & and so if there is an for which this set disappears, then the two ellipsoids do not intersect (35]. I. We can rewrite Eg as
E, = (F 1z) IF
5) where Σ, =
and J7, = (sE + (1 -1 B) Thus, the two ellipsoids centered at Ii and jB do not intersect if and only if ((s) is negative for some s E [0, 1]; i.e. &, is empty for some s. As the centers of each ellipsoid are a function of position within their respective contexts, we find that (Supplemental):
K K(S,TA,TB) =1
1
Σ
o(vN + 9)²
+
If for every pair AA BB, there is an for which this is negative, then our two thickened manifolds do not intersect. As such, we let 8" (x Tg) minimize (s,,#,, TB) for each choice of x TB. To put this condition in a similar form equation (2), we define Ø²:
1 N
(fb(zD) f (
Φ" (B) =
(6)
(=A fA(EA)
+ ( (((B))
Our condition on K(s, T.) RB) can then be written in similar form to the simpler noise model:
min (Ā"(IA, > (VN + +9)²
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Figure (A) The distributions for the minimum distance in rate space Smin (Top) and the analogue used for the rate dependent noise model, N "iin (Bottom), constructed from kemne density estimates. At large N, these approach gaussian. Plots are one dimensional rooms, with room length 1m and firing field widths W 1/3m. (B) The probability that two contexts are distinguishable as function of the number of neurons and at different noise levels, for the rate independent (Top) and dependent (Bottom) noise models.
(C) The logarithm of MNN), the number of storable contexts as function of N, at PM- = 95% confidence. In black is the predicted large N scaling, >N + 1 In
Here, 9 again accounts for the nonzero width of the noise annulus. For both models, the width of the annulus y is (11) in N, and so makes no contribution to our final results at large N. Indeed, we performed the numerical calculations for multiple values q. and find that large N our results are unchanged. In generating our figures, we use 2. Note that we are enforcing very strict definition of separability for both noise models. When separation between the two manifolds is greater than the threshold distance, the probability of confusing the two contexts is vanishingly small. Less strict conditions would still allow for good (in a practical sense) performance context discrimination, but using to
stricter definition engenders several advantages. First, the geometric approach assess capacity allows us to easily generalize to more than two contexts. Second, is important to be conservative possible in capacity calculations and such strictness should prevent overestimation of the number of decodable contexts. Finally, our approach avoids ceiling effects for changing parameters of the model; that is, by making the task as hard possible we observe impacts of changing different parameters on performance that would be hidden by performance plateaus on easier tasks.
Having these pairwise separability conditions for two contexts, we then wanted to determine how many contexts M are storable by the place cell system. To do so, we first replace our statements for particular environments and with probabilistic statements for any pair of rooms A and B generated at random. The probability that two rooms distinguishable is then the probability that the following pairwise separation conditions are true:
Constant Noise: P(2 Rooms are Separable) P( min (ZA) > 2o(vN N + 4)) (8)
Variable Noise: P(2 Rooms are Separable) = P( min No* (εA,TE) > (VN (VN+ ))) 2A.E
(9)
For notational convenience, we define òmin = min d(fa(z) (x fB(B)) (((( and dinnn = mill A.2 o² (R). The probability that two rooms are distinguishable can then be written in terms of distributions over these variables as:
20(vN
Constant Noise: Pg =
P(ōmán )dō
(10)
Variable Noise: Pg =
P(N N N
(11)
That is, we can determine P2 as long as we can calculate the distributions P(ōmin) and P( These distributions approach normal distributions for large N (Fig. 2A. Supplemental). We use
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numerical methods to find the mean and variance of these distributions. That is, we generate many pairs of rooms with unique place maps for each room and then reconstruct these underlying distributions using normalized Kernel Density Estimation (KDE) while varying the value of N (the number of neurons). Finally we can use these reconstructed distributions to calculate P2 for both noise models as function of N and the strength of the noise (Fig, 2B).
Given the above, we can estimate the total number of storable contexts, M, of the place system as a function of key parameters of the system. First, we determine how M scales with the number of neurons N. Given the probability that any pair of rooms is distinguishable, we can estimate the probability that M environments are distinguishable, PMs, via union bound:
P(M rooms are distinguishable) P(U)z) rooms i and are distinguishable) - P2
(12)
In weak to moderate noise regimes, this inequality becomes approximately saturated (Supplemental).
M(ar-1)
Thus, have Pas 2 P2
To find the number of storable contexts given N neurons, (N), we can increase M until Par falls below a desired confidence or allowable error, and call the M where this occurs the number of storable contexts. For numerically derived values, we use PM .95, but note that this only changes prefactors, and the scaling behaviour is independent of this choice. MCM
Equivalently, we can simply invert Pas Re P2 (Supplemental);
to find M(V) for a given confidence PM
lM)(PM) M(N) R V Tog( Tog(P3 P2(NN) + + -
(13)
In the limit of system dominated by noise, M (NV = 1. However, if the noise is more reasonable, we find that M scales exponentially with N for both noise models (Fig. 2C, Supplemental). Here, Y is constant that depends on firing field widths, noise, and room geometry but, critically, is independent of N at large N. We can calculate ? in terms of the distributions of òmin and N mmm as (Supplemental):
never meet our geometric constraints, and
/((mmin)/ N 20 )²
(6 mind ạ
7/8 = 2 Var [main]]
7 = (
)2
(14)
2/ Var Pmin
Here, "/8 and Yo refer to the exponents in the fixed noise model and variable noise model, respectively. One can readily show that at large N, the equations for gamma for both noise models will become independent of N (Supplemental). In this large N regime, we can then characterize the number of storable contexts solely using the mean and variance of the distributions P(6min) and P(No dn We accordingly calculated the number of distinguishable contexts numerically, and compared with the predicted large N behaviour (Fig. 2C), finding good agreement. Our results demonstrate that place coding allows encoding of exponentially many contexts with an exponent controlled by the amount of neuronal noise (Fig. 3).
2.2 A Tradc-off Between Spatial Specificity and Context Segregation
Realistic hippocampal place cells have tuning curves of varying widths. Indeed, across the dorso- ventral axis of the hippocampus, place field widths can vary by nearly an order of magnitude 21 24), with ventral place cells having wider tuning than dorsal cells. As such, we next explored how the exponent of the number of stored contexts 7 scales as function of firing field width (Fig. 3, Supplemental). To do so, we numerically reconstructed the distributions P(ōmin) ) and P(N min) for various place cell widths. In our model, increasing the widths of place field tunings starting from small sizes generally leads to increase in the distance between representations of environments.
That is because, especially in small environments, a relatively small number of place cells will show place fields, and hence small place fields lead to sparse population activity, reducing the absolute distance between firing vectors different environments. Increasing place field widths increases the average neuronal activity within an environment, thus pushing the encoding manifolds apart. As result, we expect larger fields to increase contextual capacity. fact, small environments (Im - 4m). optimal context discrimination performance occurs with firing fields that are about the size of the environment (Fig. 3). We can get sense of why this happens as follows. Consider smaller environments in which less than half of all place cells are active due to the Gamma-Poisson

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Figure 3: The calculated value of the exponential 7 at large N of equation (14), function of firing field width and neuronal noise. The environments are 1m (A and B) and 1m² (C and D). (A and C) represent the rate independent noise model (Gaussian), while (B and D) are the noise dependent model (Poisson-like). White lines demarcate the transition into non-separable regime. We find better performance for the lower dimensional environments and the Poisson-like model. For larger environments, smaller relative widths become preferable (Supplemental, Fig. 8).
When place cells have extremely large widths in this regime, each neuron either completely on or off within a given environment, and so each context becomes associated with a random binary identifier, that will be unique with high probability. Thus the environmental context can be read off simply by noting which place cells are active, although there is no location resolution at all. By contrast, in larger rooms, most place cells will have at least one firing field. So, although the sparse firing of narrow place fields makes harder to discriminate contexts based on their responses, the largest, environment-sizer firing fields also become useless in this case because essentially all cells will be active every room (Supplemental), expect tradeoff between the twin goals of context and location discrimination that depends directly on place field size, and indirectly on environment size due to the gamma-poissn statistics used to generate place field centers.
statistics of place cell activation 29 30
either case, we
Tuning the firing field widths lets us explore the trade-off between two presumed objectives for hippocampal function, encoding of position and encoding of multiple contexts. While wider fields are generally better for context segregation, as wider fields result in less variation in population level firing between locations. Thus we expect trade-off between spatial information encoded within a context, and the ability to separate contexts, tuned by the widths of place cell firing fields. To formalize this trade-off, must determine how will explicitly characterize both spatial specificity and context segregation. To characterize spatial specificity, we chose to utilize average decoding performance decoding current position & from the firing rates f(x)) of the place cell system. Naturally, good performance occurs when the decoded position typically agrees with the true position, ((t ²)) m is small at most positions. To avoid a particular choice of decoder, we invoke the Cramer-Ran bound, which lower bounds the covariance of any estimator by the inverse Fisher Information:
clear that they are not optimal for spatial specificity,
(( (t - )(r()) 2 Ix) (E, (V @ (V2))-1 ((+ ( 2 2 Tr(I='(x))
(15)
(16)
When represents position in one dimensional context, the inverse Fisher Information is scalar. If x is not one dimensional, then the inequality is statement about the difference between the covariance and the inverse of the Fisher Information being positive semi-defnitt, so that a bound on the mean squared error can be found by taking trace. In both noise models, the Fisher Information can be calculated exactly in terms of the tunings of each neuron within an environment as (Supplemental);
1 V.LA@va Is = σ
Is -- of)
V
(17)
We can now characterize spatial specificity by calculating the average spatial resolution by averaging the Cramer-Rao bound with respect both position and context. The objective for maximizing the spatial resolution can be formalized by minimizing In<(Tr[Z (=)))=²A (Fig. 4A). Tuning firing fields for high spatial resolution drives the firing field widths to minimum set by the population size. Clearly, this is at odds with the first objective for storing many contexts, which drives firing fields to be larger. Here we find our anticipated firing field width trade-off between these two objectives. The character of this trade-off depends on how formalize objective function with respect to firing
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Figure 4: (A) The Cramer Rao Bound for both noise models. (Top) represents the rate independent model, while (Bottom) represents the rate dependent model. (B-C) The optimal width as function of the relative importance between the two objectives. Using In M to characterize context decoding leads to a sharp change in the optimal width, while In P2 leads to a more gradual change. In both cases, there is trade-off between storing high resolution information and storing many contexts, tuned by firing field width.
field width W. We consider objective functions:
L X log (M W)) + X) log (og C2 - + (1- A) log(((T[T- '1/2)
(18)
(19)
Here e [o, 1] interpolates between the two objectives by setting the relative importance of each, N is the number of neurons. W is the width of the firing fields, P3 is the probability that rooms are separable given N and W, and (N, W) is the number of storable contexts (see discussion below eq. [2]. As the relative importance shifts from a high contextual capacity to high contextual resolution, optimal firing field width shrinks to a minimum set by the averaged Cramer-Rao bound. Such capacity rpsolttyresoolution trade-offss are consistent with those demonstrated in recurrent neural networks (36). For the first choice of objective function, the optimal width jumps abruptly as we vary (Fig. 4B). The second choice of objective function, on the other hand, strongly penalizes widths for which context segregation becomes impossible, leading to a smooth transition of the optimal firing field as we vary À (Fig. 4C). Regardless, we see the same clear trade-off between our two objectives for each choice of formalization. This result suggests that the difference in field size across the dorsal-ventee axis of the hippocampus may reflect segregation of coding function by optimizing for different objectives rather than just a gradient of spatial resolution as is commonly posited [17]
This is also consistent with experimental evidence that dorsal hippocampus largely recruited for spatial tasks, while ventral hippocampus typically shapes contextual response
22 23
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2.3 Field Clustering near Boundaries improves Context Segregation
So far, we have assumed that the firing field centers are uniformly distributed. In reality, place cells often drift near positions of interest and frequented locations, such as boundaries or rewards. If place cells form a compressed representation of experience, then we can reasonably propose that the density of place cells a location should reflect increased resolution for memory formation near that location. This clustering could also have an effect on context separability. Indeed, we predict that such clustering improves context segregation, and demonstrate that biasing place cells towards the boundaries of contexts can improve the ability of the place system to discriminate between them.
A uniformly distributed place cell population will, in general, have less overlapping fields near the boundaries of an environment. Since discrimination between contexts critically depends on the overlap between place cell firing fields, this lack of density by the boundaries increases the probability of confusion between contexts. This observation matches perceptual intuition. In the center of environments, animals able to reference distal cues as well different proximal cues in the

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Figure 5: (A) An example surface swept out between distances in code space by positions ZA, d(fa(x (B((B)) We have analagous surfaces for the dependent noise model. (B) The height, on average, of the minimum of the this surface for both noise models, and various firing field widths. (C) The optimal values of a derived from this approach, for both noise models and an approximation from equalizing firing ratios from the boundary and the bulk (Supplemental). (D) The sample to sample average the surface removes variations due to the random choice of fa. The minimum of this surface is near the boundaries at a 1, but jumps discontinuously the center as decreases.
environment. Adjacent to a boundary, the boundary is the dominate cue and likely obscures other cues that could potentially distinguish environments. In model, we add in-homogeneity the point process for generating place field centers to increase place field density near the boundaries. For one dimensional contexts, we can parameterize this in-homogencity via a symmetric beta distribution, 3(x L; a, a)) (Supplemental) over space. Here a acts as a 'uniformity' parameter. With a = 1 we recover the homogeneous process and have uniformly distributed place cells. Values of will progressively bias firing field centers towards the boundaries.
To understand the effect of the bias at where the distance in rate space is smallest. These distances are given by ( A), (n(nn) and K (s( TA,, can be viewed as two dimensional surfaces swept out by FA and FB (Fig. 5A), while for two dimensional contexts these could be viewed four dimensional surfaces. We take the sample to sample (annealed) average first, to find where, on average, the minimum of this surface is likely to be found (Fig. 5D). With a 1 this minimum occurs most often near the boundaries of either context A or B. As we decrease the minimum of the averaged surface near the boundaries increases until eventually jumps discontinuously the center (Fig. 5D). The value of for which this jump occurs is dependent on firing field width and the noise model under consideration, but is independent of N large (Fig. 5B). Wider firing fields tend lead larger optimal bias (Fig. 5C). In two dimensions, considered distribution of firing field centers that is product of beta distributions, (ax I,; Ozz, 0z) (yy T.; Ciys a.y). The analysis for the x direction and y direction separate, which leads to identical optimal values for the uniformity parameter a in the 1-D) case.
look discrimination of pairs of contexts A and B,
for each noise model, respectively. For dimensional contexts, these
Discussion
Many researchers have proposed that the place system in the neural substrate of the cognitive map and that global place cell remapping plays a critical role in storing information about environmental Further, recent work may implicate the role place maps general, which might explain the need for such large
context [1 15 16
short term memory formation contextual capacity. In this work, we have built upon these proposals by explicitly demonstrating under realistic firing statistics that the place cell system's 's context storage capacity grows exponentially
37
38
39
40

---

with the number of neurons, and by calculating the associated exponents. This large capacity is consistent with the notion that the hippocampus capable of pattern separating context and encoding many experiences [41 4 and here we demonstrate that a place-like coding scheme alone is sufficient in this regard. To achieve this result, we developed a geometric model of place cell activity, which allowed us to explore how this capacity changes as a function of the number of place cells in the system and of place cell firing field properties. While our strict conditions on pairwise separability leads to coding scheme that is robust to noise, we note that less strict conditions may be more realistic, and better suit an animals behavioural needs. We primarily focused on global remapping here, but conjecture that the qualitative structure of our results remain unchanged by including the effects of partial remapping. Including these effects will give each manifold an additional width along& few dimensions due to variations that are not due to neural noise, but rather due to partial or rate remapping. Additionally, we have not considered here the complexity of decoders of the hippocampus. Although we show that context separation is achievable, the requirement of simple decoding, as well as the architecture of the underlying hippocampal network, will further constrain the contextual capacity [31 32 (3).
We then explored implications of this model as it pertains to various objectives of the hippocampal code. In particular, we revealed a trade-off between precise encoding of local position and discrim- ination between different contexts. We found that tuning individual place cells for encoding local position leads to smaller place field widths, while increasing place field widths leads to improved per- formance for context discrimination. The size of place fields increases from the dorsal hippocampus to the ventral hippocampus (1 24]. and suggest that this mixed population of neurons allows the hippocampus to perform both objectives efficiently. That is, our model suggests that place cells of the dorsal hippocampus are better tuned for fine grained memory, while the more widely tuned ventral cells are better tuned for pattern separation and storage of many contexts. This is consistent with experimental evidence, in which dorsal lesion typically impair spatial memory, while ventral lesions do not. Conversely, ventral lesions have been demonstrated to impair contextual memory, for example decreasing response in contextual fear experiments, but have minimal affect on spatial tasks [22 23 24
25 26).
We also found that biasing place cell centers to cluster near environmental borders improves context discrimination. Over-representation of place field activity near boundaries is well documented [27], and we predict that this bias will systematically vary across the dorsal-ventral axis of the hippocampus, with the more widely tuned ventral place cells displaying greater bias than dorsal place cells. As rodents typically explore near boundaries of environment, the need for higher spatial resolution in these locations may also lead to a similar bias. In fact it is well established that developmental and self-organization mechanisms can produce efficient structural and functional optimizations (vision:
48 49 : olfaction: (50 51]: spatial cognition: (52)) and here we are suggesting that similar processes may operate in the place system.
46]; audition: 47
While we have explored hippocampal codes in isolation, interactions with other spatially tuned cells, such a egocentric and allocentric border cells, likely have effect this bias not explored here, suggesting yet another intricate interaction between allocentric-egocentiic representations in the hippocampus (53). If place cells are implicated for general episodic memory, such interaction may imply that boundary cells play a role in general memory. Exploring this interaction is topic for future work, and our approach provides a foundation for exploring these avenues.
Finally, we have also focused on physical one and two dimensional contexts in this work, but our geometric formulation generalizes to higher dimensional and abstract spaces. Our derivation of the exponential scaling is independent of the dimension, and so we predict that the hippocampus should also be able to segregate context and distinguish locations in more abstract spaces efficiently. It is worth noting however that there is still an appreciable drop in performance when moving from one to two dimensional spaces, and so the system is likely incentivized encode abstract spaces with lower dimensional structures when possible.
Acknowledgments: We thank Dori Derdikman, Genela Morris, and Shai Abramson for many illuminating discussions in the course of this work, which was supported in part by NIH CRCNS grant 1 12544-155444001 and by the NSF and DoD OUSD (R&E) under Agreement PHY-2229929 (The NSF AI Institute for Artificial and Natural Intelligence). VB was supported in part by the Eastman Professorship at Balliol College, Oxford.
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A Supplemental
A.1
Simulation Distributions and Firing Fields
A.1.1
Gamma-Poise Poinssn Distributions
The number of firing fields each place cell has is determined by a gamma- gamma-poisson distribution, as described in 22 30]. This is mixed poisson model, with gamma distribution acting the mixing distribution. That is, the rate variable of the poisson model is gamma distributed with parameters o, B. We
2.25 in and two dimensions, respectively. We use 4m/ I. and 3 8m2 2 /A one and two dimensions, respectively. These create distributions consistent with those seen in experiment, below 29 [30]. The total number of neurons for generating these distributions was 200
1.5 and
0.5
0.5
03
of
0.4
0.3
0.10
0.2
0.1
0.05
0.0
0.0
0.0
0.00
Number Fields over m
Number Fields over m
Number Fields over 4
Number of Field over
0.8
0.4
0.6
0.5
Percent Celts
Percent of Calls
0.3
0.4
of
0.2
0.3
0.2
0.1
0.0
0.0
0.00
Number Fields 2.m
Figure The number firing fields per neuron that arise from the gamma-poisson distribution. Top is one dimensional rooms that vary in length from 1m to 8m. Bottom is two dimensional rooms that vary in size from 1m2 Sm? As rooms get larger, neurons are recruited for representations more often
Under these statistics, most neurons are silent in small rooms, so two contexts will share few active neurons. In large rooms, few neurons have firing fields, and two contexts will share many active neurons.
A.1.2 Field Definitions
For each randomly generated room, each neuron, indexed by j. has a, fields drawn from the gamma- poisson distribution described above. Each firing field center is placed according to a homogencous (first two parts) or nonhomogencous (last part) poisson process over space. Firing fields gaussian, so that the total spatial tuning is given by a sum of gaussians:
aj
fa)( 1H + CAAA E exp 2w 2)2 (T PA.))
(20)
All neurons given base firing rate of 21 and Cj.A is chosen that active neurons have firing rate of 20H2 for simplicity. Max firing rates are not drawn from a distribution, to speed up pdf convergence times. Likewise, widths not chosen stochastically, and have equal fixed width. This also speeds up pdf convergence times, and additionally lets us study the effect of changing firing field widths in isolation.
A.1.3 High Dimensional Gaussians
Here we outline standard method of demonstrating the exponential localization of gaussians to shells in high dimensions B3 34). If z is distributed according normal N dimensional spherical
14

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gaussian with variance o², and R = |z/2 is its norm, then the distribution of R is
p R
R
N// exp(
(21)
20
where Sy is the area of the N dimensional unit sphere. Setting aR PR)(R) = O, we find this distribution has maximum at o N.N. Now for any oq << ovN, we have that PR(OVN + tgg = Sx(ovN + oq) lov
(22)
(2TO²)
ON
8 (N-1)))
(23)
(2
SN * -PVカ- (NN----NN-
(24)
(2TO2)N/2"
d²
(x(oov))xpp(-oa
(N 11)ln((1 }}
(25)
VN
pnnoovN) ((NN))xp((-VV -VN q² (N 1)(
q x²
2N ((NN)3/3))
(26)
VN
= -Px(ovN)cxpp-pp>> - ONN-1/2)
(27)
And so the mass of high dimension gaussian is exponentially localized to an annulus of radius VVN. We can choose q such that the majority of the probability mass of z is within a a sphere of radius a (vN + q). For all the figures in the main text, we use q = 2, which leads R 99.8% of the mass of p(z) being within a sphere radius ovvN + g). As q is of order 1 in N, choosing larger values leads to negligible effect derived values at large N, which we additionally verified numerically.
For the variable noise model, & ~ NO, of exponentially localized to an ellipsoid with major axis of length v (ofA((A)) Including the annulus width, we have that the ellipsoid width in a particular direction is ( + q).
We have then that & at any position I'A is
A.1.4 Ellipsoid Intersection
We want to check the manifolds widened out by noise intersect. For the simple noise model, checking if the spheres centered at (x(x)) and (B( for some pair EA.TB equivalent to checking the distance between their centers is greater than twice their radius, or d(fx(EA), fD(EB) >
(v(VN + q). For the variable noise model, we need an efficient way to check if two ellipsoids intersect.
From 35). two ellipsoids high dimensional euclidean space, defined by centers pas UB and covariance matrices A,, EB intersect if and only if the convex function
((s) 1 shbe AMM (1 - 3) p MeEe
(28)
becomes negative for any choice of E [0, 1]. The proof is given in (35. Here,
Σ, = ~Σ + (1 =)
(29)
In (sEAA. (1 B)bB)))
(30)
In the Poisson-lie noise model, we check if the ellipsoids centered at fA(#A), fB(B)) in rate space intersect. These ellipsoids given by:
(r fa(=))) +
<
(31)
(r - ((=) + -0)
<
(32)
Here, (T( ( will drop the J and Fy for conciseness, with the understanding that there is an implicit dependence. To check for intersection, we plug these into the form of the convex function K(s):
K ((,, (SB SA) / (VN
POEA + - -
'SB fA)
15

---

The ellispoids centered at (T(A) (Z() do not intersect if and only if there exists an s* E [0 such that (s",, , AA) [35 Rearranging, we can write this as:
E (e((n)) (X(aa)) (
X(B = Φ(vN + q)
The choice that of that minimizes ((s, AA, IB) changes with To. and so we can let s (TA,TB) represent this value. We define o (IA,TB) by
1 N
(fn(zB) fa(=a))²
s" ( =
(33)
(= ((asss)
This definition lets us write:
(() = d(VN q)2 60 (IA)
(34)
Now we want ellipsoids centered at fa(xA), fL(TB) to not intersect for any TA. TB- which amounts requiring that K(s (T) TA maXa E E (ss TB), TA, TB) < 0. To put this in a similar form as the gaussian case, we can write this as the requirement that
for all TA or equivalently
min No (I), > (VN + 2)² SA.BB
(35)
A.2 Gaussian Distributed Miminum Distances
A.2.1
Gaussian Model
We define 8min min ( 8min is function of the maps fA, fa, which chosen at random with respect to a Gamma-Poisson distribution. Here, we find how the mean and scale of 8min scale with N.
The distance in rate space evaluated at some pair positions XA, XB for large will look like the square root of the sum of the square of many random variables, as the firing fields are randomly chosen. particular, each term in the sum itself is independent of N, so to get the rough shape as function of N of this distribution, can discard some of the finer details of fA, fB.
N d(fa(Xa))f((nn)) = -CA(xN)- fx(()))
(36)
The construction of the firing fields of each neuron occurs independently, so Y; (fA(XA) fig (Xx))² represent set of i.i.d random variables with some mean variance which is independent of N, allowing us to use the central limit theorem. We can write then
N E(s(xA) -UA(XA)-(((Xm))
(((a(X)),(((xm)
(37)
N + Z1
(38)
N / 1 Zz
(39)
Where Aá E|Y)), and Z, is normally distributed random variable with mean zero and variance
N NVar Y.). by the central limit theorem, and Z2 is normally distributed with zero mean and variance
Yar As nı is independent of N, large N, we can expand the square root around small Ze:
d(fa(X)) ~ / Nia + 7
(40)
We find then that the distance in rate space between XA and X is distributed normally at large N, with mean scaling like VN and variance that is independent of N,
Ni VN, x
16

---

The constants are not determined here, though will be calculated in simpler case later, and fit to simulations. Now ïás and \ may depend on the particular choice of positions XA and XB, but this will still hold positions where the minimum is most likely to be found, and so P(mmin) will also have a mean that grows with /N and unit variance,
8min ~
(41)
for yet determined constants.
A.2.2 Rate dependent model
We also need to calculate the behavior for P(NØ*) large N. We start with the definition of ø², and want to show that it is gaussian distributed
N Σ
(B((z)) (((()))((()A))) (RA 17 S
(42)
Again, consider any particular fixed XA, Xp. Then the content of the some random variable with respect to the distributions on fA. fB. The elements of the sum have non-zero mean and variance and are i.i.d. This is because in constructing neural firing fields, the placement of fields for neurons and j are assumed to be independent. By the central limit theorem, have then that
be viewed just
N6 No (XA, XB) N, N2N)
(43)
or equivalently
0" ("AA. Ni Xg/N)
(44)
for some constants jo,, Ag- The values of ja,, A may depend on the exact position but will be independent of N sufficiently large N, and so in particular, fluctuations of 6 will mostly be due to position. This implies that, as before, the minimum value N will have a distribution with the same scaling as No* any position, but with different constants.
N Nomin ~ (N X2,N
(45)
8
E[8mma// 1:
1.5
min)
1.0
Nel
2
250
250
Number of Neurons N
Number of Neurons
10
Ac [Var
VariNo
3.0
8
2.0
1.5
250
250
Number of Neurons
Number of Neurons
Figure 7: Numerical demonstration that the constants μø, As H&g As are independent of N for large N in 1-d, as predicted.
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A.3
Union Bound and Product Ansatz
In the main text, we assume that the probability that M contexts are distinguishable is approximately
a function of the probability that any pair is, via a saturation of the union bound Pas = P2 More transparendly, assume:
P(M Contexts are distinguishable) R
Pli and j are distinguishable)- = P2 (MM (46)
(6)
Strictly speaking this is a union bound PM S. P2 (M11222 as the probability that rooms 1 and 2 are distinguishable and rooms 2 and 3 are is not independent. We argue that this bound is approximately saturated at large if noise is not too strong, by demonstrating each context occupies a vanishingly small relative volume in rate space N gets large. This argument is rough reason think that the ansatz is true, but not an exact proof.
We consider the rate independent noise model to start. The total available volume in rate space is F. m How we have incorporated noise lets neuron firing rates extend past Fman: as it max set before noise, and so firing rates can "spill out" of this N dimensional box. For one dimensional
rooms, fA carves out curve of length 1= f s dx Σx 2
in rate space. For D dimensional rooms,
LA defines surface, which has area
S / d v/ det
(47)
Here g= = JJ with Jf being the jacobian Jj = 0z, In D) dimensional rooms, g will be matrix, with each element scaling no faster than linearly in N. The determinant then can scale no faster than D/2 so at worst, we expect the surface area to scale in N like
S D/2
(48)
where c is some constant. To go from the surfaces carved out by the maps f to the volumes occupied by the firing rates of the neurons, need to consider the effect of adding noise. Now we can roughly approximate the effect noise by putting an N - D spherical cross section to each point along the surface. We avoid exact integrals as we primarily care about the scaling. The radius of each cross section is oVN, /N, and each cross section has volume:
π N/2
VN VN-D(OVN)
(OVNNN-D
I +1)
Or, using stirlings approximation to get the scaling in N,
VN-D(OVN)
2ne
N/2 D/2 N)
(49)
V(N D)r N
D
(2mee
2
N
N=D
(50)
VNN-D)r D)
N N-D
The approximate volume in rate space of the thickened manifolds then scales no faster in N than
V, x c D/2
V(N D)m (2
1
N
N
2
(51)
N N-D D
Assuming the dimension of the environment D is small (for physical environments, D is 1, 2, or 3, but we might be interested higher dimensional abstract 'environments') compared to N and N large, then at worst the scaling is
V, x PTJ (2iTed2) Nop
(52)
We have neglected the volume of the "caps" of these manifolds. For example, the one dimensional rooms define a tube in rate space, and have half-spherical caps on each end. The volume of the N dimensional sphere of radius VNo² is 1 (2iee²)NN/2 N/2 In higher dimensions, this additional
VN=
18

---

component would scale very roughly like the length of the boundary of the surface S times the volume of the N D sphere.
The proportion of the volume of the thickened manifold to the total volume of rate space, V, /FN then vanishing in N provided that the neurons are not too noisy. We get approximate condition of the form V2Teo < Fmaax- A more precise calculation might improve deteriorate this bound, but interestingly this mimics typical conditions on signal to noise ratios. Each room occupies a vanishingly small fraction of the total volume in rate space N grows large. Similar arguments hold for the ellipsoidal noise model, as each ellipsoid occupies smaller volume than its osculating sphere.
maz
A.4 Exponentially Many Rooms
Here, we find the scaling of the number of rooms M with respect to the number of neurons N. We will use the product ansatz:
PM = P2 (M 1)/2
We invert this equation by taking the logarithm of both side, then solving the quadratic formula for M, to find that
(og(Pss) 1
M(N)= V log((2(()) + 1/2
(53)
Here, we can let PM represent a confidence. For example, if we are okay with being 95% percent confident storing multiple rooms, then (MV) becomes equation purely dependent on (2V) Now (N) is, at large N, equal to an integral of gaussian for both noise models. We start with the simpler noise model. We have that
PB P(6 > 2VNN +9))
(54)
1 11 F-F(22(VN N +1)
(55)
Φ( 20 N +g) N
(56)
As
Here, F represents the cumulative distribution, and & is the cumulative distribution of the standard normal distribution. The approximation comes from the central limit theorem at large N, with 148 = B(ōmin) 1/ /N and X = Var(ómin) are both independent of N. This can be written in terms of the error function for the first noise model
(2) + (1 erf( (0(VN 9) MSVN
(57)
V2As
VN -20(VN +1)1]
(1 + erf(
(58)
V2As
We can bring the second noise model into similar form:
P2 P(PPpmmmmm min No (E*, > (VN RAiEn
2))
(59)
1 F((VN +2)
(60)
1 D (VN + 2) (
Nuo
(61)
AOVN
(VN + 2P NuA))
2 (1 erf(
(62)
V2AOVN
Nus - (VN + 1)
(1 erf(
(63)
V2AOVN
Here, F is the cumulative distribution for d'min N, MS B(d //N and X3 Var(d both independent of N.
)/N are
19

---

In both noise models, we can write = two noise models, respectively. P2 approaches either I for large N, depending on the strength of the noise with respect to the mean distance μr. We have then:
= (1 + erf(u)), with given in equation 58 and 63 for the
lMg(PM)
M(N) = V22 1og()(1 erf(u))) 4 + 1/2
(64)
In the noisy regime, this approaches large N. In the weak to moderate noise regime, we have that u approaches & as N approaches in both models, and the emor function approaches 1. We are interested in its asymptotic behaviour. We can use following expansion at large u:
erfu- =
(u
2 2 -3 + ...)
(6.5)
VT
Expanding to leading order u then gives:
log PM
M(N)= = 2
(66)
V
log(1
1
exp
uu + (u-
2
V
Now we can expand the logarithm by noticing that for small values,
A In(1
A (
+ +
(67)
x)
So to leading order, we have:
M(N) ~
-2
(68)
log (1
2
1
exp
(-u
u
1 exp (-1²)
v 2og 1/
1-122
(69)
V
= 2/ VT log Pm Vuez
(70)
At large N, the constant q is negligible, and so drop it, giving for the rate independent and the rate dependent model as u noise models gives
He 2n VN, and ns /N, respectively. Plugging in for both vZAa
VIAs
In(1/Pa)) ps 20
N8T exp 1/| 16 140
20 ) A
Model 1: M(N)
(71)
As
M6 (8TTN)1/4 exp 1 11a=- Model 2: M(N) In(1/Pa))
16 )
(72)
X
2o
In both noise models, the number of storable contexts as function of scales like
M(N)
2
(73)
with the constant 7 in each noise model being given by
E[omin /N - 20 )2 76-
E c min
④
To 2. N Var
(74)
2 Var(Amin
These values are shown for various noise levels, firing field widths, and room sizes the main text figure and the supplemental figure 8.
20

---

Polsson Moice fau (Pa-)
80
Polcicnn Moice (P>1)
80
80
80
50
Noice of
Exponent Velue:
Gouss Noise
20
80
20
40
10
0.75
0.50
Width WLL(1-0) (1-D)
Width WIL (1-D)
Width (2-D)
Width W.L. (2-D)
Gaussaan Noise (122))
Polecon Notse fivud (Ht2l)
Polccon Notte froe (Mtrl)
50
30
60
60
Bppnnen Value
Gaussaan Noise
20
80
20
20
20
10
20
0.50
0.75
0.25
0.50
0.25
0.50
0.75
0.25
Wiath WIL (2-DI
Wieth WIL (1-DI
Width WIL (2-D)
Wieth W/L (2-D)
Gauss Neise (822)
Polocon Noise fraug (M22)
Polccon Noice frau (H21)
30
Eo
30
60
0.6
Exponent Valuee
Gauss Noise
20
20
20
20
20
20
0.25
0.50
0.75
0.25
0.50
0.75
0.25
0.50
0.75
0.25
0.50
0.75
Width WLL (1-DI
Width W/L (1-DI
Width WIL (2-D)
Width WIL (2-DI
Gaussian Nosse (K22)
Polsson Noise frcug (H2²)
Polocon Noise fooug (K22)
30
30
0.6
80
Exponent Valuee,
Gaussian Noise
20
20
20
20
20
20
0.25
0,50
0.75
0.25
0.50
0.75
0.25
0.50
0.75
0.25
0.50
0.75
Wiath WIL (3-DI
Wiath WIL (1-DI
Width WL (2-D)
Wieth W/L (2-D)
Goussian Noise ot (62²))
Poloson Noise fruug !Hz²))
Coursian Noise (Hzz")
Polozon Naise frued (Ho2")
30
30
60
0.4
20
80
20
40
0.3
10
10
Ro
0.10
0:25
0.50 0.75 Width WLL (1-D)
0.25 0.50 0.75
0.25
0.50 0.75
0.25 Width WIL (2-D)
0.50
0.75
0.0
Width WLL (1-D)
Width WLL (2-D)
Figure 8: The value the exponential yy at large as a function of firing field width and noise. From left to right are 1d rooms, Gaussian noise; 1d rooms, Poisson-like noise; 2d rooms, Gaussian noise; and 2d rooms, Poisson-like noise. Rooms increase in size from top to bottom (1m, 2m, 4m, 8m, and 16m rooms, and 1m? 2m² 4m2 Sm?, and 16m² rooms). Under Poisson Gamma statistics for the number of active fields, we expect in smaller rooms, where the number of active fields small, to preference large fields for the objective of context segragation. Conversely, in large rooms, extremely large fields become harmful for context segregation. White demarcates the region where decoding multiple contexts possible. The Poisson-like model is generally much more robust noise.
21

---

A.4.1 Infinite Width Derivations
For neurons with infinite width, the number of storable contexts is purely a function of the number of neurons which have active fields. In this limit, there are N neurons, each context gets a binary identifier. Consider two contexts A and B. If each neuron has probability 0 of being active, the distribution the hamming distance between these two contexts given by
P(H = k) = 1 (m²) NN
[e + (1 1)))) - 0)1*
(75)
Now we are interested in P(min ((ff(xA)))))) (I)), fB (EB)) > 20 In the infinite width limit, the spatial depedence is lost, and + Smax H. We can write then that the probability that two rooms are separable is
46²
P(H
N)
Σ
P(H
(76)
max
k=
4
N1
The distribution P(H) is well approximated by a gaussian at large N, with mean μ. = 21 - 1) and variance o² 21 0)0[(1 - 0)² + 0). From this, we get an integral similar to the one arrived at in the main text. At large N, the probability that two rooms are separable approaches one if 20(1
0)
10a Rooms are optimally separable in the infinite width limit when 1/2.
A.5
The Cramer-Rao Bound
To quantify a population level code for position, we consider estimators of position constructed from population level firing, and compare estimator 4 at a particular position is bounded below by the inverse Fisher information:
position. In one dimension, the quality of any
(( - x) )r|2 = Var($) 2 I (x)
(77)
In one dimension, the Fisher metrie is given by:
I I= E 2 dr 1)3]
(78)
Where the log likelihood of firing rates given position. In higher dimensions, this Cramer-Ree Rao bound is a statement about covariance:
( r) x) 2 (x)
(79)
Here, 2 indicates that the difference of the LH and RH side is positive semi-definite. To get the mean-squared error, we need take trace over the inverse Fisher information metrie:
(( - 2) = Tr ((+ - =) - x)
(80)
Tr Cov($) 2 Tr(I-
(81)
In higher dimensions, Cov(&) > I Ix:) is statement about the difference being positive semi- definite, which leads to our trace condition above. The Fisher Metric is given by:
I nE, (VV.))
VV. V
(82)
The averages in the Cramer- Rao bound are taken with respect to firing rates given positions. As we are interested in having good spatial resolution across positions and contexts, we additionally take averages over both. Below, we plug in the log- liclliooss for both noise models.
A.5.1 Gaussian Noise
In dimension, we have that
=- 202 -
1= Σ In P(rilz)=-
Σ
- fx()))²
(83)
22

---

The Fisher Information then is given by
I = nE,,(( a
1)²1
(84)
Or
= o.
E E Σ (T) - fa(o)) E - f((2))
(85)
We have that
- +0502 i - fi(z)
(86)
So the fisher information is
0 1)
I
nE,
(87)
Ox
1 Br
- +1OA))))))))))))
(88)
(1) - -2) +
(89)
σ2 Σ [3)
(90)
The CR bound will be small when this is large. This implies that, the wider the firing fields, the smaller the derivatives of l will be at most locations, which leads to a smaller Fisher Information, and so a worse bound. This leads to typically narrow fields. However, if we are in region where all firing fields are zero, the FI vanishes, and the bound explodes. This implies that the widths shouldn's t be so small that there are regions with no firing fields. We also have that as the noise increase, the information decreases, and the bound gets worse, as we expect.
In 2d, replace derivatives with divergences, and take trace the end. We have:
- Σ
20 (ri f2(x))²
(91)
1 02 (n. (n==((()VVAG
(92)
Going through the same steps as before, we find that we just replace the ordinary product from before with outer product:
I 1 V
(93)
A.5.2 Rate Dependent Noise In the second noise model,
1 d/
P(rv(z)
exp (0))(())
(r: f2(z))²
(94)
Our log likelihood is
(rr f2(x))) In 2m0b((()))
(95)
2Φ12(r)
201 (r-- - fx(x))² 1 In fx()) +...
1
(96)
2
23

---

This time, we will start with divergences, and demote them for 1d rooms. We have:
1
1
V
45 (ri ff + 20f22(11 (ri
(97)
2014
2FA
(fa(2)² (z) + V
(98)
(x)
The Fisher Information then
E + +5 - -?) + (z) - r3)
IV V
(99)
(z)
(x)
We have that
Er f1(x)) + 0f2(1)
(100)
E, (r Iktt = (f)(z) (((6))+0f(())))))) +0
(101)
Er = -fx(x)* =
(102)
= 20000000022100000000000000010000
(103)
E [r = -C000---00000000000-000000000010
(104)
(105)
Most terms in the expectation cancel, except for where = j:
I (z) 20 f (x)
V 8 V
(106)
(2)
VIA V
(107)
(2)
2
In 1-d, we replace the outer product and the divergences with ordinary scalar product and derivatives, respectively.
A.6 Biasing Towards Boundaries
We investigated the effect of biasing place cell centers towards the boundary of environments. In 1-d, we bias the place cell centers toward boundaries by drawing centers from a symmetric beta distribution, that the bias can be characterized the parameter family of distributions:
I((a) T(a)² (n/L)a-1(1 p/L)a-1
P(p) B(//L;
(108)
We expect an improvement in decoding multiple contexts when the total squared activity near the bulk and near the center is, on average, equal within a context. In dimension, for environment of length L, we want then approximate equality:
f((0))a S((/2))
(109)
The average is with respect number of firing fields centers and their location. If assume for simplicity that each neuron has exactly one firing field, then we have equality when
(
),, ( + (u-
(110)
Taking the average with respect the beta distribution, the optimal bias can be found then by finding where the following integral is zero:
T(2x))
r(a)2 S'' dp(n/L)o-1(1 (1 -0- - -
(111)
Or.
[ dxx ae) 412
222 (0--12)) =0
(112)
24

---

2.00
1.0
Dist of Centers, 8ixiL. a. ()
1.75
1.50
0.8
1.25
0.7
1.00
0.6
0.75
0.5
0.50
0.4
0.25
0.3
0.00
0.25
0.50
1.00
Position x/L
Figure 9: Beta distribution, from which we draw place cell centers. a is degree of uniformity, with reproducing the uniform distribution.
To make this calculation numerically stable, we do the integral over an interior region (small c)
dxr--1 (1 x²) 2-)a-1
(w- (x-1/21²
(113)
We can invert this numerically to get an approximation of the optimal based on equalization of firing densities in the bulk and at the boundary. We find decent agreement between this rough calculation and the true optimal bias found in the main text for both noise models, slightly overestimating and underestimating the rate independent and the rate dependent model, respectively (Fig 5 C of the Main Text).
A.7 Additional Computational Details
To reconstruct probability distributions for distances, we generate large number of pairs of rooms with fixed number of neurons and widths, calculate distances in rate space, and then find the one dimensional distributions over distances by taking kernel density estimate over these generated distances. Although the space of possible curves in our rate space large, since we only need to reconstruct the dimensional distributions in 6min and Omin which are asymptotically gaussian, sampling this space suffices. We create between 1000 and 10000 samples in each numerical experi- ment. To speed up calculations, then use scikit learns's built in Kernel density estimator with a gaussian kernel to reproduce the distributions in &min and Φmin [55]. As the reconstructed distribution can depend heavily the choice of bandwidth used for the kernel, we calculate the kernel density multiple times for various choices of the bandwidth, and pick the best bandwidth via 5-fold cross validation. All code run on a machine with an Intel Xcon-2 245 processor and with Titan Xp GPU for parallelized workflows.
perform the sampling of surfaces in parallel using PyTorch (54).
25

---

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Reviews:
[{"review_id": "f2f0dftlfx", "reviewer": "Reviewer_FBYZ", "summary": "In this paper, the authors take a geometric approach of analysing context-encoding capacities admitted by place cells population firing. Specifically, through examining the manifolds underlying neural activities within different environments, the authors propose to quantify the separability of context encoding based on the overlap between the manifolds. Somewhat surprisingly, the authors noted a tradeoff between spatial specificity and contextual separation. Under the context separation constraints, the resulting place cells are tuned to be densely distributed around boundaries, which is a useful testable experimental prediction.", "questions": "See weaknesses.", "limitations": "See weaknesses.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper is well written, with clear pointers to mathematical details where appropriate.\n- The tradeoff between spatial specificity and context separability is novel and sounding.\n- The proposed geometric analysis is a novel framework for studying the nature of contextual representations in place cells.", "weaknesses": "- The paper only addresses global remapping, and did not study the implications of proposed model in terms of partial or rate remapping.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the comments and for taking the time to review our paper. Indeed, we chose to focus our paper on global remapping, with only brief discussion of rate and partial remapping. This is because global remapping has a more dramatic and constraining effect than rate/partial remapping in our analysis: in terms of our manifold picture, partial and rate remapping alter or deform the existing manifolds, while global remapping involves “jumping” from one manifold to another. Global remapping often occurs in situations where context shifts dramatically, which for example can occur when animals move, or are moved, from one environment to another, even if the environments are superficially similar (Leutgeb et. al., 2005 Science, Alme et. al. 2014, PNAS). Partial and rate remapping often occur when an environment is modified, such as via slight changes to wall geometry, introduction of olfactory cues, or movement of cue cards within the same environment (Leutgeb et. al. 2005 Science, Bostock et. al. 1991 Hippocampus, Anderson and Jeffrey, 2003 JNeurosci). In our view, these experiments demonstrate that partial and rate remapping involve a “deformation” of existing memory tasks, while global remapping occurs and is required when completely new memory tasks arise – such as entering a new environment or changing the animal’s goals – so that the animal is required to remember both the current task and the past ones separately, as opposed to slightly modifying the old task. In the context of our manifold picture, partial and rate remapping involve the deformation/refinement of the neural manifolds we are considering, while global remapping involves switching from one manifold to another.  Our results about the capacity for storing context follow from estimating the number of such manifolds that can be packed into the activity space in the presence of noise.  We can account for partial and rate remapping also in our framework by giving each manifold an additional width due to variations in the encoded structures that are not due to neural noise, but rather due to variations that arise from partial remapping. Thus, the qualitative structure of our results will remain unchanged by including the effects of partial remapping. We will discuss this extension in the revised submission – thank you for encouraging us to do so.  Note that, mechanistically, in the context of a network implementation, partial/rate remapping might involve the alteration of some continuous attractor implemented by the hippocampus, while global remapping would involve jumping from one continuous attractor to another.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JHpg6jS8LL", "reviewer": "Reviewer_V9KM", "summary": "This paper offers a computational investigation on the problem of encoding environmental information using population codes based on place cells, which are known to play a key role in hippocampal encoding of context, experience / goals and spatial locations. The authors propose to analyze the geometry of hippocampal codes, with the aim of precisely quantifying the capacity and properties of context encoding by place cells with different firing properties. Their analysis reveals that the number of storable contexts (i.e., strictly separable manifolds) grows exponentially with the number of place cells, showing that the hippocampus might in fact have an exponential storing capacity under realistic firing statistics of place cells.", "questions": "-\tPlease always explicitly describe the variables used in equations (also in figure captions).\n-\t“To determine whether two contexts manifolds are separable, we use a strict criterion: the two manifolds are separable if and only if they do not have any intersections.”. This requirement seems quite strong, because it assumes that we need to decode context from any position in the manifold… Could it be replaced by a smoother criterion and/or by some form of probabilistic (linear) discriminability?\n-\t“We postulate that selective inhibition along the hippocampus will lead to different types of memory impairment for spatial tasks”. How would be possible to test this hypothesis experimentally? The authors mention “performing confusion experiments” but it would be interesting to better discuss how such experiments would look like.\n-\tPlease always explicitly state where the information can be found in the Supplemental material.\n-\tLine 157: determine", "limitations": "The authors have discussed possible limitations of their work, though not in a totally explicit way.", "rating": 7, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "I think that this work is a nice example of a theoretical contribution in the field of computational neuroscience. I am not familiar enough with the related literature to evaluate the originality of the approach, but from my understanding the analyses are well-conducted and well-motivated. The paper is written in a clear way.", "weaknesses": "The main issue with this submission, according to my non-expert opinion, is that it might have a limited relevance (and impact) on the NeurIPS community. Indeed, although I am aware that NeurIPS welcomes contributions more focused on neuroscientific aspects of neuronal computation, there is not a single NeurIPS paper cited in the literature, suggesting that this type of work might be more appropriate for other venues.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We appreciate the time taken by the reviewer to review our submission, and the suggestions provided.\n\nWe believe that our submission is relevant to the NeurIPS community, and in particular, to the neural coding section of the Neuroscience topic, which is listed in the call for submissions. As contextual discriminability and spatial memory are both implicated by hippocampal function, approaching both through the geometry of the underlying hippocampal codes will be of interest to this community. Although the work presented here does not directly touch on the applications or theory of artificial networks, we believe that a better understanding of biological neural networks will lead to a better understanding of artificial networks, and give insights into how to design them for certain tasks. For instance, research at DeepMind (Banino et. al, 2018, Nature), by Cueva and Wei (ICLR 2018)  and by Sorscher et al (NeurIPS 2019) has found that grid-like representations, similar to those found in the Medial Entorhinal Cortex, emerge naturally for networks trained on spatial navigation, leading to deep learning agents with mammal-like navigational abilities. Likewise, place cells in the hippocampus are implicated in general short term memory (Benna and Fusi, 2020, PNAS). So a better theoretical understanding of hippocampal function will be of interest to the wider NeurIPS community when it comes to designing networks capable of flexible memory storage and retrieval.\n\t\nThe reviewer asked if we could have used alternate criteria for separation of neural manifolds. Indeed, there are some options. For example, the work of SueYeon Chung involves a linear separation criterion for perceptual manifolds in deep neural networks (Chung et. al., 2018 APS, Chung et. al, 2020, Nature Communications). Pursuing a simpler separability criteria between activity manifolds, like the linear separation of point-cloud manifolds used in the above work, but in the context of hippocampal coding, is a possible direction we would like to explore in future research.\n\t\nAs for selective inhibition experiments, these are possible via induced lesions to various regions of the hippocampus in rodents.  Lesions in the dorsal region of the hippocampus should lead to impairment on tasks in which high spatial resolution is crucial, such as during maze navigation, while lesions to the ventral hippocampus should lead to greater impairment in contextual tasks. (A typo in our text that inverted dorsal and ventral, but we will correct this in the final version.  Also see comments and responses to reviewer Dte5.) Some of these experiments have already been performed (again see comments to Dte5), and are in agreement with our (typo-corrected) predictions. We will also expand on the possible confusion experiments that could be run to test our hypothesis, and include the relevant citations. With regard to explicit descriptions for variable names and references to the supplemental material, we will make these both more clear in the final submission as well.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "hBlRndqMxn", "reviewer": "Reviewer_DtE5", "summary": "This paper develops an analytical framework for understanding the coding of space and context in the hippocampus. The novel contributions include a characterization of how tuning width contributes to these functions and the trade-off between them. In particular, smaller tuning widths improve spatial localization but impair context discrimination. The authors suggest that this explains the functional role played by the gradient of tuning widths along the dorsal-ventral axis of the hippocampus. The model also explains why place cells might cluster near boundaries.", "questions": "- Can the authors do a better job relating their work to existing literature (see Weaknesses)? I understand that due to space limitations it is unlikely that they will be able to comprehensively address this literature, but I want to make sure that the work is at least largely in alignment with what is known.\n\n- The predictions depend on noise level. Is this something that can be tested experimentally using firing rate variability?", "limitations": "The authors briefly discuss some modeling limitations. There are no negative societal effects of this work.", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 4, "strengths": "- The paper addresses an important set of issues within systems neuroscience in a novel way. The hippocampus has long been implicated in both spatial and contextual coding, but the relationship between these has not been elucidated so systematically. I believe this could have a potentially large impact, at least within the community of theorists.\n\n- The paper is clearly written.\n\n- The analysis is rigorous.\n\n- The paper makes some interesting experimental predictions (though see below for connection to existing experimental work).", "weaknesses": "Overall my critical comments are relatively minor (see below). I do have one major comment pertaining to the model's empirical predictions. The paper makes an interesting and testable prediction that the more widely tuned cells in the dorsal hippocampus are specialized for context discrimination, whereas the more narrowly tuned cells in the ventral hippocampus are specialized for fine-grained spatial discrimination. First, I want to point out that this is backwards: ventral cells have wider tuning than dorsal cells. The classic study of this is Kjelstrup et al. (2008, Science), not cited here (see also Komorowski et al., 2013, Journal of Neuroscience). Oddly, the authors cite two papers to support their claim about the dorsal-ventral axis (Lee et al., 2020; Tanni et al., 2022), neither of which actually support this claim. The Lee et al. paper only measures activity in dorsal cells, and it's not clear which subregion was measured in the Tanni paper.\n\nThe authors propose selective inhibition experiments to test these predictions. In fact, such experiments have been done, and unfortunately they don't consistently support the predictions (none of the studies mentioned below are cited in the paper). The model would make more sense in light of at least some of these studies if the dorsal/ventral division of labor was reversed from what the authors proposed, consistent with the electrophysiology data. The review by Fanselow & Dong (2010, Neuron) provides a more systematic discussion of studies dissociating dorsal and ventral subregions.\n- Richmond et al. (1999, Behavioral Neuroscience) showed that ventral lesions actually *improve* water-maze performance (a classic test of fine-grained spatial memory), whereas dorsal lesions apparently have no effect. Ventral lesions also impaired contextual fear conditioning, but dorsal lesions apparently had no effect except in the test phase where they *increased* conditioned responding to context. See also Bannerman et al. (2003, Behavioural Brain Research) for related results.\n- Hock & Bunsey (1998, Journal of Neuroscience) showed that dorsal, but not ventral, lesions impaired performance on a spatial delayed alternation task, which requires memory of actions in particular spatial locations.\n- On the other hand, Moser et al. (1995, PNAS) showed that dorsal lesions selectively impair spatial memory, whereas ventral lesions do not.\n\nThe authors predict that place cells should cluster near boundaries to support context segregation. No studies are cited to support this prediction, but this is something that has been studied. Consistent with the model prediction, place fields tend to occur near boundaries (e.g., Wiener et al., 1989, Journal of Neuroscience; Hetherington & Shapiro, 1997, Behavioral Neuroscience).\n\nMinor:\n\n- \"hippocampus\" is inconsistently upper-case/lower-case. I think it should be lower-case.\n\n- p. 3: Eq. 1 should have brackets around the exponentiated term.\n\n- p. 5: \"dominate\" -> \"dominant\" [also p. 8]\n\n- p. 6: \"an increase the distance\" -> \"an increase in the distance\"\n\n- p. 6: \"severally\" -> \"severely\"\n\n- p. 9: \"environmnet\" -> \"environment\"", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We are grateful for the positive response, as well as the valuable suggestions and pointers to existing literature.\n\nIn the initial submission, there was a  typo that propagated throughout our text, exchanging dorsal and ventral, and hence also the predicted scaling along the dorso-ventral axis. We thank the reviewer for pointing this out! In fact the ventral cells have wider tuning then the dorsal cells, and we will fix this typo in the final paper, and include citations to the Kjelstrup et al. and the Komorowski et al papers.  Note that our predictions are based more on the relative firing field sizes of place cells, and not on where these cells are located, so the key results of our paper remain unchanged. In particular, we predict that cells with wider tunings are better tuned for contextual separation but are worse at fine tuned spatial tasks, while those with narrower tunings are better tuned for tasks associated with fine-grained memory, but worse at contextual separation (though both are still able to contribute to both tasks). Correcting our dorso-ventral typo, our suggestions about experiments involving the dorsal hippocampus should be replaced with experiments involving the ventral hippocampus, and vice-versa. In particular, as the ventral cells are those with wider tuning, the corrected prediction is that selective inhibition of ventral cells should lead to worse contextual performance. Conversely, the fact that the more dorsal cells have narrower tuning suggests that selective inhibition of the dorsal hippocampus should lead to worse performance in fine grained memory tasks under our model. After this correction, our model is in line with the experimental papers you mention. In particular, our theory is supported by the result that lesions of the dorsal hippocampus impair spatial tasks (Moser et. al. 1995, Hock and Bunsey 1998) while lesions of the ventral hippocampus do not meaningfully affect performance on spatial tasks, and we will cite these experiments in the final paper. Likewise, the fact that conditioned contextual fear responses, like those described in Richmond et. al. (1999) and Bannerman et. al. (2003), are impaired after ventral lesions also supports our hypothesis that the more widely tuned neurons are better for context separation. \n\nAs for the predictions involving clustering near boundaries, we will likewise cite the relevant experimental work in the final paper. In particular, the increased incidence of place cells near boundaries in Wiener et al. (1989) is in line with the predictions made by our model. As for noise variability experiments, one could inject white noise into hippocampal neurons through electrical input, or pharmacologically increase firing variability, which under our model should reduce both spatial and contextual specificity. In particular, there is a noise threshold above which the ability to separate context disappears in our geometric model. However our condition for contextual separability is likely stricter than one implemented by the hippocampus of a realistic animal, so contextual separation should persist to some extent past this noise threshold – i.e., we expect the threshold to be more of a smooth crossover than a sharp transition. \n\nFinally we will address the minor edits suggested by the reviewer.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper develops a theory for understanding the tradeoffs between encoding space and encoding context in the hippocampus. The novel contributions include a characterization of how tuning width contributes to these functions and the trade-off between them. The authors use this theory to explain a number of experimentally-reported but not-well-theoretically-understood facts, including the functional role played by the gradient of tuning widths along the dorsal-ventral axis of the hippocampus, and why place cells might cluster near boundaries. The results and the theoretical concepts are well explained, broadly accessible and potentially impactful for computational and empirical neuroscience.
Decision: Accept (oral)