Integration and multiplexing of positional and contextual information by the hippocampal network

The hippocampus is known to store cognitive representations, or maps, that encode both positional and contextual information, critical for episodic memories and functional behavior. How path integration and contextual cues are dynamically combined and processed by the hippocampus to maintain these representations accurate over time remains unclear. To answer this question, we propose a two-way data analysis and modeling approach to CA3 multi-electrode recordings of a moving rat submitted to rapid changes of contextual (light) cues, triggering back-and-forth instabitilies between two cognitive representations (“teleportation” experiment of Jezek et al). We develop a dual neural activity decoder, capable of independently identifying the recalled cognitive map at high temporal resolution (comparable to theta cycle) and the position of the rodent given a map. Remarkably, position can be reconstructed at any time with an accuracy comparable to fixed-context periods, even during highly unstable periods. These findings provide evidence for the capability of the hippocampal neural activity to maintain an accurate encoding of spatial and contextual variables, while one of these variables undergoes rapid changes independently of the other. To explain this result we introduce an attractor neural network model for the hippocampal activity that process inputs from external cues and the path integrator. Our model allows us to make predictions on the frequency of the cognitive map instability, its duration, and the detailed nature of the place-cell population activity, which are validated by a further analysis of the data. Our work therefore sheds light on the mechanisms by which the hippocampal network achieves and updates multi-dimensional neural representations from various input streams.


Effective two-state model for hippocampal CANN activity
We show below how the two-state model pictured in Main Text, Fig. 3B, can be derived from the definition of the microscopic CANN model, see Main Text, Methods. The dynamical evolution of the CANN ensures that the log probability of a configuration of activity s = {s i } is given by [1] where r if the rodent position and L * is a constant term such that the sum of the probabilities e L over all 2 N configurations s is normalized to unity. For simplicity, we consider that the bump of activity consists of a×N active neurons s i = 1 with place-field centers as close as possible in a map, say, m = A, where a is the fraction of active neurons in any time bin. This corresponds to the limiting case of zero neural noise, β → ∞ [2]; calculation of effective potentials at finite β is much more involved and requires the use of sophisticated statistical physics techniques able to take into account the fluctuations of neural activities, see [3].
Let us define r bump as the radius of the bump, i.e. the maximal distance in environment m between the rodent position r and the place-field centers r m i of active neurons. We have where δ 2 is the elementary portion of surface per place cell, defined as the total area of the environment, L 2 , over the number of place cells, N . We thus obtain the expression of the bump radius as a function of the activity, r max (a) = L a π . (3) Contributions to log-likelihood due to inputs.
We assume that the CANN has activity localized in map m, and that the whole system is in the conflicting phase, with P I = A and V = B. The contributions to L due to the visual (V ) and path-integrator (P I) inputs reads, up to quadratic terms in a, where γ = γ P I if m = A and γ = γ V if m = B. According to the definition of the bump radius r max , we have dr r e −r 2 /(2σ 2 ) = γ 1 − e −r 2 max /(2σ 2 ) .
Contributions to log-likelihood due to recurrent connections.
We now consider the contribution L recurrent to the log-likelihood coming from the recurrent connection in the CANN. The coupling J ij between neurons i and j is the sum of one interaction specific to map A and another one specific to map B, see Eqn. (9) in Main Text, Methods. Assuming again that the bump of activity is localized in map m = A, we neglect the contribution to L due to the interaction specific to map B. This simplifying approximation amounts to an error of the order of a 2 , see [2] for more details. We obtain where I 0 is the first kind modified Bessel function of zero order.
Case of mixed state.
Assume now that a fraction α of the bump is localized in map m = A and the remaining fraction, 1 − α, is localized in map B. The log-likelihood of this mixed state is obtained by summing the expressions of the log-likelihood in state A above with activity a → α a and of the log-likelihood in state B above with activity a → (1 − α) a. The result is shown in Supplementary Fig. A. As indicated in Main Text, the amplitude γ J of the recurrent connections controls the depth of the well separating the two complete bump states (all A or all B), while the ratio γ P I /γ V controls the asymmetry of the log-likelihood profile and favors one of the two states.

Effects of parameters on the model properties
The CANN model is defined up to a set of parameters: (a) the level of neural noise in the simulated activity, β; higher β corresponding to lower noise. This parameter is formally equivalent to the inverse temperature in the Monte Carlo simulation; (b) the strength of the recurrent connectivity, γ J ; (c) the strength of the two inputs, γ V and γ P I ; (d) the spread of place fields and positional inputs, σ; (e) the number of neurons, N .
(f) the mean activity (fraction of active neurons at any time), a.
A fully-detailed analysis of the response of the system to the each of these parameters is beyond the scope of this paper, and previous works have fully characterized the behavior of the model in the absence of positional inputs [1][2][3]. Hereafter, we show how some of these parameters control the dynamical properties of the flickering of the cognitive map and the ability to navigate, i.e. the correct positioning of the bump of activity in the position defined by the V and P I inputs.
These two quantities are indeed observable in the CA3 electrophysiology data, through the map and position-decoding analysis. A characterization of their parametric dependence in the model is therefore a necessary step to a correct quantitative modelling. For this reason, we will here divide the parameters into two classes: • the structural parameters, N , σ, a. The number of neurons N was varied from a few hundreds to a few thoousands in simulations. To keep the contributions to the total in put H i,t acting on neuron i at time t independent of N , we scale the recurrent connection strength γ J as 1/N , see Eqn. (9) in Main Text. This ensures that the sum of local inputs over all active neurons due to these connections has a finite, fixed value as N grows. This is why we will compare below the value of γ J × N to the other input strengths, γ V and γ P I . In addition, we have fixed the average linear size of place fields to σ/L ∼ 0.125, which sets the average area occupied by a place field to 2π(σ/L) 2 10% of the environment total area, a value comparable to experimental findings [4]. The average activity (in a time bin) was fixed to a = 10% throughout our simulations to match the values fixed in previous works focusing on the same model in the absence of inputs, see discussions in [2,3].
• the control parameters (γ J , γ V , γ P I , β), that have a predictable influence on the behaviors we are interested in. Note that the four control parameters are redundant, as the properties of the model depend only on (β × γ J , β × γ P I , β × γ V ); we may therefore fix one of them and let the other three vary. We now study how the model properties depend on the values of these parameters.

Navigation of the environment
The model is explicitly designed to mimic the representation of self-location in the hippocampal network under the influence of positional inputs. A natural question is how the values of parameters influence the capability of the model to actually represent the correct position in a single map, that is, the correct centering of the neural bump around the input position. Consider the case of coherent inputs at a certain time t, i.e. PI and V point to the same position r t in the same map , and let us assume that the bump is correctly centered around r t . As the input position changes in the next simulated time bin t + ∆t, P I and V will try to activate place cells corresponding to a shifted location, effectively pushing the bump to r t+∆t . If the positional input is too weak compared to the recurrent network connections are too strong, the bump will fail to update to the new position, being trapped by the strong connection with the active cells at position r t . Similarly, a very high value of β, i.e. a low neural noise, would have the effect of enhancing the roughness of the energy landscape, in the positional space, and of trapping the bump and imparing its motion. As a consequence, the model would lose the ability to correctly navigate the environment. Conversely, a very low value of β would result in the inability of the model to condensate the bump of activity [3], therefore losing any notion of represented position. The inverse of the mean positional error t can be used as a proxy for the navigation ability, and is shown in Supplementary Fig. B as a function of β and of the relative strength γ V /(N γ J ) (in the balanced case γ V = γ P I ). The navigable region (yellow) has a triangular shape that widens with higher values of the input strength, meaning that the temperature has to be fine tuned for low values of γ V /P I , while it can take a wider set of values in the presence of strong inputs.

Flickering of the cognitive map
As discussed above, the system acts as an effective two-state model when the two inputs are put into conflict, i.e. point to the same position in different cognitive maps. The transition of the bump from one map to the other happens stochastically, and its dynamical properties are controlled by the parameters of the model. Characterizing this dynamics in the simulated test experiment is rather involved, since the positional inputs move at a variable speed (we use the recorded trajectory of the real rat as input) and a fast change of the positional input can facilitate the evaporation of the bump from one map, increasing the transition rate between maps. We here analyze the dependence of two data-testable quantities on the parameters. The first is the statistics of permanence in the visual-cue associated map or in the PI-associated map during the conflicting phase, as a function of the relative strength between the two inputs, shown in Supplementary Fig. C. We see that a ration γ V /γ P I close to 1 results in a mean fraction of flickers (MFF) close to 0.55. This value, slightly different from the expected 0.5 is due to the inertia of the bump that, for few bins after the teleportation, tends to stay in the PI-associated map. Since each simulation is carried for a finite number of time bins after the teleportation (600), this discrepancy is explained as a consequence of the finite-time simulated for each trial.
Next we analyze a dynamical quantity, i.e. the mean sojourn time of the activity in one of the two maps, given a balanced value of γ P I = γ V , see Methods for the definition of the sojourn time. This quantity is directly proportional to the height of the barrier described in the two-state approximation, which is controlled by the network connectivity strength γ J and the parameter β.
High noise (small β) or weak connections (low γ J ) is expected to enhance the probabiity of crossing the barrier easy, and to make the sojourn times low. This statement is confirmed by the results shown in the diagram in Supplementary Fig. D.
Putting together the results reported in Supplementary Figs

Relationship between sojourn time and correlation time
where τ A and τ B are the mean sojourn times in, respectively, map A and B. A straightforward calculation shows that the time correlation C(τ ) between the map state at times t and t + τ is decreases exponentially with the delay τ only, with an average time equal to Hence, the correlation time τ 0 is approximately given by the smaller mean sojourn time among τ A and τ B . The distribution of sojourn times in each map for experimental CA3 data [5] is shown in Supplementary Fig. E. Regions of consecutive theta bins whose decoded representation disagree with the external light conditions are marked as "PI" regions. Vicevesa, if they agree with light conditions, they are marked as "Visual". Results obtained after application of our map decoder to the recorded CA3 data of [5]. As shown in the bar plot and from the ANOVA comparison, the permanence times in the two maps during the conflicting phase have roughly the same distribution. Results obtained with map-decoding threshold L0 = 2.3.
4 Inference of path-integrator realignment times -discussion on parameters p 0 and p e To identify the realignment times of the PI we first introduce a simple probabilistic model for the hippocampal representation to be incoherent with the light-cue conditions (flickering time bin) as a function of time elapsed after the switch (see Main Text Methods). This procedure needs an input value for p 0 , the probability of flickering during the conflicting phase, whose consistency can be checked a posteriori by computing the mean flickering frequency in the conflicting phase. In Supplementary Fig. F

Assessment of performances of map decoder
Our map decoder (based on the inference of an Ising model for each cognitive map) does not use any information about the current rodent position. Its performance can be assessed against the correlation-based decoders used in [5] (which compares the activity s t to the expected activities in maps A and B at the given rat position) by means of the classical binary-classifier theory [6][7][8][9]. The Ising model was shown, first on retinal ganglion cell recordings, and, more recently, on prefrontal cortex [10,11] and hippocampal data [12,13], to provide a good approximation for the distribution of population activity configurations. The performance in the decoding task has been shown to be superior to rate-based decoders on CA1 data [12]. The standard tool used to compute the performance of binary decoders is the Receiver Operating Characteristic (ROC) diagram [6]. This diagram is drawn by computing the true positive rate (TPR) and false positive rate (FPR) as a function of different thresholding values, and plotting the resulting curve in the TPR-FPR plane. These quantities can be defined in the context of our map decoder as follows: for each theta bin t the decoder outputs a value ∆L(t), which is then interpreted as referring to map A or B depending on its value compared to a moving threshold Θ. Note that this is slightly different from the map-decoding method reported in Main Text, since it does not allow undecoded statistically-not-significant bins.
To match the vocables used in the ROC framework we will arbitrarily follow the convention that the output is positive if the map is decoded to be A, and negative if the map is predicted to be B. Doing so, a True Positive is defined as a correctly-decoded environment A (with respect to the light conditions: m t = A = light cues), while a True Negative will be a correctly-decoded environment B. The final observable (area under the ROC curve) is symmetrical under the inversion of this convention, which is summarized in Table 1. The decoding capability is finally assessed by applying the decoder to two "constant" test sessions, where the environment is constantly set to A and B, respectively. Assuming that the neural representation is stable under fixed light conditions, we can compute the TPR and FPR of the decoder by counting how many theta bins are correctly and falsely decoded in the two reference sessions. For a specific value of the threshold Θ, this corresponds to a point in the FPR-TPR plane. By varying this value we then draw the curve as the succession of the corresponding TPR-FPR values. The standard quantitative measure of the decoding performances is the Area Under the Curve (AUC) of the ROC diagram [6]. According to this measure, the ideal decoder has AUC = 1, while random guessing would give AUC = 0.5. All the decoders, tested on constant test sessions, i.e. where no teleportation is performed, show very high performances, see Supplementary Fig. H. Note, in addition, that our functional-network based decoder is robust against the presence of correlations between the maps: it shows much better performance than correlation-based methods for CA1 recordings, where maps are much less orthogonal than in CA3 [12].  Table 1: Denominations used for the four possible events, depending on the output of the decoder and on the environment-defining cue. The cue is not changed throughout the reference session.