{"title": "Coordinated hippocampal-entorhinal replay as structural inference", "book": "Advances in Neural Information Processing Systems", "page_first": 1731, "page_last": 1743, "abstract": "Constructing and maintaining useful representations of sensory experience is essential for reasoning about ones environment. High-level associative (topological) maps can be useful for efficient planning and are easily constructed from experience. Conversely, embedding new experiences within a metric structure allows them to be integrated with existing ones and novel associations to be implicitly inferred. Neurobiologically, the synaptic associations between hippocampal place cells and entorhinal grid cells are thought to represent associative and metric structures, respectively. Learning the place-grid cell associations can therefore be interpreted as learning a mapping between these two spaces. Here, we show how this map could be constructed by probabilistic message-passing through the hippocampal-entorhinal system, where messages are scheduled to reduce the propagation of redundant information. We propose that this offline inference corresponds to coordinated hippocampal-entorhinal replay during sharp wave ripples. Our results also suggest that the metric map will contain local distortions that reflect the inferred structure of the environment according to associative experience, explaining observed grid deformations.", "full_text": "Coordinated hippocampal-entorhinal replay as\n\nstructural inference\n\nTalfan Evans\n\nInstitute of Cognitive Neuroscience\n\nUniversity College London\n\ntalfan.evans.13@ucl.ac.uk\n\nAbstract\n\nNeil Burgess\n\nInstitute of Cognitive Neuroscience\n\nUniversity College London\nn.burgess@ucl.ac.uk\n\nConstructing and maintaining useful representations of sensory experience is es-\nsential for reasoning about ones environment. High-level associative (topological)\nmaps can be useful for ef\ufb01cient planning and are easily constructed from experience.\nConversely, embedding new experiences within a metric structure allows them to\nbe integrated with existing ones and novel associations to be implicitly inferred.\nNeurobiologically, the synaptic associations between hippocampal place cells and\nentorhinal grid cells are thought to represent associative and metric structures,\nrespectively. Learning the place-grid cell associations can therefore be interpreted\nas learning a mapping between these two spaces. Here, we show how this map\ncould be constructed by probabilistic message-passing through the hippocampal-\nentorhinal system, where messages are scheduled to reduce the propagation of\nredundant information. We propose that this of\ufb02ine inference corresponds to co-\nordinated hippocampal-entorhinal replay during sharp wave ripples. Our results\nalso suggest that the metric map will contain local distortions that re\ufb02ect the in-\nferred structure of the environment according to associative experience, explaining\nobserved grid deformations.\n\n1\n\nIntroduction\n\nLocalizing in an environment relies on two sources of information. Firstly, unique sensory inputs\nmay indicate absolute location in space. Secondly, path integration (PI) can update previous location\non a metric map by integrating self-motion. Sensory inputs are required to correct the accumulation\nof error by PI, but problems arise when their role in localization occurs simultaneously with learning\nof their correspondence to locations on the metric map (SLAM) [10]. In general, computing the joint\nmap-location distribution requires probabilistic inference over previous sensory observations and\nmovements given their respective uncertainties. Associative representations can be computationally\ncheaper when used to perform high-level planning [56]. However, organizing associative structure\nin a metric space allows for ef\ufb01cient integration of new experience and the inference of metric\nrelationships between sensory states in the absence of physical experience. This \u2018short-cutting\u2019 ability\nis crucial for ef\ufb01cient exploration and navigation [58; 55].\n\n1.1 Place and grid cells\n\nNeurobiologically, grid cells (GC) in the medial entorhinal cortex (mEC), whose \ufb01ring \ufb01elds are\narranged on a periodic hexagonal lattice in space, are thought to play a role in PI [17] and constitute\na metric map of space [22]. Their \ufb01ring patterns are stable over time suggesting stabilization by\nenvironmental cues [24; 12]. Conversely, place cells (PC) in the hippocampus (HPC) \ufb01re at distinct\nlocations [43] and are thought to respond to speci\ufb01c sensory stimuli such as environmental geometry\n[44; 30]. PCs represent states in sensory space such that their activity most often re\ufb02ects the animal\u2019s\n\n33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada.\n\n\fcurrent location, their synaptic associations constitute an associative map of an environment [40]\nand their connections to GCs stabilize the GC metric map. Although PC and GC activity most often\nrepresents the current location, coordinated sequential \u2018replay\u2019 of remote cells (i.e. whose \ufb01ring \ufb01elds\nare non-local) also occurs [15; 45].\n\n1.2 Summary of contributions\n\nThis work proposes a novel dual-systems account of probabilistic localization and learning in the\nHPC-mEC system, on both an algorithmic and implementation level [33]. Predictions from our\nhypothesis are evaluated by comparison to existing experimental data by both numerical simulations\nand theoretical analyses.\nWe propose that the HPC-mEC system operates in two distinct regimes. When navigating using\na known map (i.e. locations of sensory states in metric space), an online system probabilistically\nintegrates PI and sensory information for localization (Fig. 4A). A simple learning mechanism allows\nthe online system to learn initial priors over the map structure (Fig. 4B). However, con\ufb02icts in the\nPI and sensory estimates of location necessitate more complex of\ufb02ine inference to correct the map,\nwhich requires inference over previous sensory observations and movements (Fig. 2) [48].\nWe show how this of\ufb02ine system can use the associative structure stored in the recurrent CA3 synapses\nbetween PCs to construct and correct a metric map stored in the synaptic associations between GCs\nand PCs, corresponding to one-shot learning. The distribution over landmark locations is computed\nvia message passing [47] between PCs. Scheduling messages to minimize the propagation of\nredundant information not only improves performance, but also produces structured reactivations\nof PCs resembling those observed during hippocampal \u2018replay\u2019 [15; 9; 50]. Our model provides\nboth a functional and mechanistic interpretation for observations of coordinated HPC-mEC replay\n[45; 64] and makes novel experimental predictions. In contrast to reward-based interpretations [36],\nour model poses replay as structured coordinated information transfer though complementary metric\nand associative representations of the world. Sharp-wave ripples [7; 42] which coincide with replay\nevents may correspond to structural prediction errors.\nLastly, when the learned associative structure is non-Euclidean, organization within a metric space\npredicts recently observed local distortions in GC \ufb01ring patterns, such that the underlying structure\nrepresented by GCs re\ufb02ects the informational \u2018similarity\u2019 of distinct locations in stimulus space.\n\n2 Related work\n\nPrevious theoretical models have proposed how current location might be represented in the population\n\ufb01ring rates of grid cells [35], updated by PI information and corrected by sensory inputs [24; 17; 13].\nHowever, these mechanisms do not describe how uncertainty in either the PI or sensory inputs could\nbe integrated probabilistically, instead assuming that input from familiar landmarks \u2019reset\u2018 the current\ndistribution.\nProbabilistic localization assumes an existing mapping from sensory stimuli to location in metric\nspace. Although several studies have demonstrated how this mapping might be learned [39; 38; 41],\nthese models do not link experimentally observed local distortions [23; 54] in the \ufb01ring patterns\nof grid cells to non-uniformities in their underlying stimulus input, or to non-uniform behavioural\nsampling.\nNeither do these models link to the phenomenon of coordinated HPC/mEC replay. During replay,\nplace and grid cells with overlapping spatial \ufb01ring \ufb01elds are observed to reactivate during of\ufb02ine\nstates (such as sleep, grooming or pausing at choice points), in spatial sequences that recapitulate\nbehavioural trajectories experienced by the animal during online behaviour [16; 9]. Forward [9] and\nreverse [16] is thought to be associated with planning and consolidation during reward-based learning,\nrespectively. Of existing models, only one provides normative insights [37] and none account for\ncoordinated mEC-HPC replay [29; 46; 64]. Ours is the \ufb01rst model to implicate replay in probabilistic\nlearning of spatial structure, providing an alternative view to reward-based accounts.\nOur model does not tackle the problem of learning the form of the metric mEC space into which the\nenvironmental structure is embedded, although it is possible that these representations can emerge\n\n2\n\n\fFigure 1: A The online model. The GC \ufb01ring rates at time t are updated by PI before correction by\nweighted input from PC \ufb01ring. Each hexagon de\ufb01nes a single grid module with NG GCs. Plotting\nspikes from a single GC against the position of the animal generates the GC \ufb01ring pattern. B Learning\ncorrects the observation model towards the predicted estimate.\n\nfrom unsupervised learning in navigational tasks [2; 62] or as the eigendecomposition of the transition\nmatrix between states in an environment [52] or as predictive functions of sensory inputs [61].\n\n3 Model\n\n3.1 Grid and place cells\n\nA single GC will \ufb01re periodically at the vertices of a triangular lattice in 2D space (Fig. 1A). GCs\nexist in anatomical \u2018modules\u2019, groups of GCs whose \ufb01ring patterns share the same spatial scale\n(distance between vertices) and orientation relative to the environment, but differ in their spatial\noffsets [22]. Moreover, the spatial scale increases in discrete \u2018jumps\u2019 along one anatomical axis of\nthe mEC, suggesting that these modules encode a hierarchical representation of space [3; 14; 34].\nG(x) describes the probability distribution over current location within a periodic, discretized region\nof state space x. Biophysically, this would be represented by the \ufb01ring rates of NG GCs G (i.e.\na discretization over the support of G(x)). Although we only consider a single grid scale, our\nresults naturally extend to multi-scale architectures, theoretically allowing encoding of ranges up to\n[21; 34; 60] or beyond [14] the largest grid scale.\nPC \ufb01ring P represents the probability of the presence of speci\ufb01c sensory stimuli, which in a spatial\ncontext can be considered as \u2018landmarks\u2019 whose location in physical space is denoted by \u00b5p. In\nt \u223c\nour simulations, the \ufb01ring of each of the NP PCs is described by a Gaussian receptive \ufb01eld pp\nf (\u02c6x(cid:48)\nPCI) is a noisy estimate of the current position in physical space\nx(cid:48). We will use the notation Bp(x) to denote the continuous distribution over the location of landmark\np in metric (GC) space (its belief ). Biophysically however, this would be encoded in the synaptic\nassociations between PC p and the GCs, i.e. the pth column of the matrix B \u2208 RNP \u00d7NG. We consider\nthese synaptic associations to constitute the metric embedding of sensory experience (Fig. 4A).\n\nPCI), where \u02c6x(cid:48)\n\nt \u223c N (x(cid:48)\n\nt|\u00b5p, \u03c32\n\nt, \u03c32\n\n3.2 Online localization and learning\n\nGiven a suitable representation of uncertainty and a known map, localization is achieved by a\nprocess of recursive Bayesian estimation (RBE), where a model-based prediction based on perceived\nmovement is corrected by incoming sensory information (Fig. 1A).\n\n(cid:90)\n\nMovement update The location distribution (grid module activity) from the previous time-step\nG(xt\u22121) is updated according to perceived movement given a transition model T (xt|xt\u22121, \u02c6ut):\n\nG(cid:48)(xt) =\n\n(1)\nwhere G(cid:48)(xt) is the movement estimate and \u02c6ut \u223c N (ut, \u03c32\nPIuI) is the noisy perceived movement\nat time t, where \u03c3PI scales the noise with distance travelled. Since the \ufb01ring of GCs are periodic\nacross space, we use a wrapped Gaussian function de\ufb01ned over the triangular lattice to account for\n\nT (xt|xt\u22121, \u02c6ut) \u00b7 G(xt\u22121)dxt\u22121\n\n3\n\n\fFigure 2: The of\ufb02ine model. A Inferred distance is a function of the \u2018overlap\u2019 in the receptive \ufb01elds.\nB Inferred pairwise distances between place \ufb01elds. C Inferred distances are used to recover the\nabsolute structure of the world. D Structural inference on static structures with noisy initial priors\n(\"Initial\"). Inferred structure is sensitive to the topology of the environment (\"Broken Ring\").\n\nthe probability of having transitioned from any of the in\ufb01nite grid tilings:\nf (xt \u2212 xt\u22121|\u02c6ut + cmn, \u03c32\n\nT (xt|xt\u22121, \u02c6ut) =\n\nPI\u02c6utI)\n\n(2)\n\n\u221e(cid:88)\n\nm,n=\u2212\u221e\n\nwhere and cmn = 2\u03bb(mv1 + nv2) is a spatial offset of scale \u03bb given the lattice basis vectors\nv1 = [cos(\u03c6), sin(\u03c6)] and v2 = [cos(\u03c6 + \u03c0/3), sin(\u03c6 + \u03c0/3)] and \u03c6 is the global orientation of the\ngrid pattern. Where grid space is represented discretely by the \ufb01ring rates of a population of GCs, the\nperiodic form of the transition function can be replaced by multiplication by a velocity dependent\ncirculant matrix T(\u02c6ut) (see Appendix C.1), linking to the eigendecomposition of diffusive transition\nmatrices [51] and generalizing a previous mechanism to the case of noisy PI [5].\n\nObservation update The predicted estimate is re\ufb01ned by incoming sensory input to give the\nintegrated estimate G(xt):\n\nwhere H(Pt|xt) =(cid:80)\n\nG(xt) =\n\n(3)\nt Bp(xt) is the observation model de\ufb01ning the likelihood of the current\npp\nsensory inputs Pt given the predicted location. The normalization constant Kt is the sum over the\ncurrent GC activity, implemented by a simple inhibitory feedback circuit: \u03c4 dG(x)\n, where E = 1 is a constant excitatory drive such that the sum of the GC activity sums to unity at\nsteady-state.\n\ndt = \u2212(cid:82) G(x)dx + E\n\np=1:NP\n\nH(pt|xt) \u00b7 G(cid:48)(xt)\n\n1\nKt\n\nOnline learning as prior formation Where the distribution over landmark locations are encoded\nin the PC-GC synaptic weights matrix B and the predicted estimate of location by the GC \ufb01ring rates\nt, a simple error-based learning rule with learning rate \u03b1 = 1e \u2212 4 minimizes the error between the\nG(cid:48)\nobservation and movement models (Fig. 1B):\n\n1\n\u03b1\n\ndB\ndt\n\n= 2p(cid:62)\n\nt (G(cid:48)\n\nt \u2212 PtB)\n\n(4)\n\n3.3 Of\ufb02ine message passing for probabilistic structural inference\n\nDuring exploration of a novel environment, the online model produces stable learning when PI noise\nis low and the transition structure is static (Fig. 1A). However, all learning is local: only the synaptic\nweights of the currently active cells are modi\ufb01ed at each time-step. This is not a full solution to the\nSLAM problem, which requires \ufb01nding the most likely con\ufb01guration of sensory observations (land-\nmarks) {bp}p=1:NP and current location (in grid space) given all historic observations and perceived\nmovements, described by the joint map-location distribution p(xt,{bp}p=1:NP |P0:t, \u02c6u0:t, x0) (see\nAppendix Fig. 1 for a summary of the anatomical mapping). Computing this requires integrating\nover all possible con\ufb01gurations of PC locations, which requires inference over previous and non-local\nobservations. There are several advantages of a system capable of propagating information through\nnon-local locations. Firstly, updates to the perceived location of a given landmark cause associated\nlandmarks to also be updated without needing to be re-visited. Secondly, multiple weak (high\nvariance) observations can together form strong hypotheses if those observations are consistent.\n\n4\n\n\fThe hippocampus as a cognitive graph The structure of an environment can be inferred from\npairwise distance observations between landmarks [10; 40]. Intuitively, consider a \u2018spring network\u2019 of\nconnected landmarks, where the edges represent noisy pairwise observations with stiffness and length\nequal to the certainty and estimated pairwise distance, respectively (see Appendix D.1). Convergence\nis contingent on the fact that, despite large absolute errors in landmark location (due to noisy PI),\nerrors in relative pairwise distance measurements are correlated such that their variance decreases\nover time [10]. Relaxing the \u2018spring mesh\u2019 is equivalent to \ufb01nding the maximally likely con\ufb01guration\nof landmarks, if pairwise distance observations (pairwise potentials \u03c8ij) are described by Gaussians\nwith mean dij and variance \u03c3ij = \u03c3P C + dij\u03c3P I that are equal to and proportional to the perceived\ndistance, respectively (the latter re\ufb02ecting accumulation of PI noise in Eq. 2; Fig. 2C; Appendix D.1).\nThe PC-GC synaptic associations can then be viewed as priors over the locations of each landmark\nin metric space, \u2018anchoring\u2019 the inferred structure which would otherwise be translation / rotation\ninvariant. Together, the associative structure and metric mapping, encoded in the PC-PC (A) and\nPC-GC (B) associations respectively, de\ufb01ne the posterior distribution over the landmark locations bi:\n\n(cid:89)\nm,n=\u2212\u221e exp(cid:0) \u2212 1\nwhere the \u03c8ij(bi, bj) = \u03c8ji(bj, bi) =(cid:80)\u221e\n\nP ({bp}p=1:NP ) = 2\n\n(cid:89)\n\n1\u2264i\u2264NP\n\ni\u2264j\u2264NP\n\n\u03c8ij(bi, bj)\n2 \u03c3\u22122\n\nij (dij \u2212 ||bi \u2212 bj + cmn||2)2(cid:1) terms\n\n1\u2264i\u2264NP\n\nBi(bi)\n\n(5)\n\nde\ufb01ne the pairwise potentials between PCs and \u03bb is the grid scale. Note that Bp(bp) here de\ufb01nes\nthe continuous distribution of the location of PC p in metric (GC) space for consistency with the\nliterature; in reality it is a discrete vector described by the pth row of B.\n\n(cid:89)\n\n1\n\u03b1\n\ndAij\ndt\n\nAssociative encoding in the hippocampus We propose that these pairwise distance measurements\nare encoded in the recurrent synaptic associations between CA3 PCs, constituting an associative\nrepresentation of the structure of space [40]. Given Gaussian place \ufb01elds, a simple modi\ufb01ed Hebbian\nlearning rule with constant decay learns the pairwise PC weights (associative map) A:\n\nwhich converges in the steady-state to Aij =(cid:112)< pi(t), pj(t) >, the square root of the correlation\n\n= pi(t)pj(t) \u2212 A2\n\nbetween the \ufb01ring of two PCs (see Appendix D.1 for more details on the choice of learning rule).\nWhere all place \ufb01elds have equivalent receptive \ufb01eld covariance, the inferred Euclidean distance of\nPC j from the perspective of i is then proportional to the true distance given a simple transformation:\nij = \u2212log(Aij) = (\u00b5i \u2212 \u00b5j)2/2\u03c32\nPC. The resulting form for the recovered distance is also scaled\nd2\nby the receptive \ufb01elds\u2019 variance (the Bhattacharyya distance) [4], such that \u2018closeness\u2019 is related\nalso to the \u2018discriminability\u2019 (Fig. 2A). We discuss this scaling constant later (see also Appendix\nD.3). Our approach differs subtly from typical graph-based SLAM systems [31; 57] which treat each\nobservation independently. Instead, the CA3 synapses effectively average over multiple pairwise\nmeasurements. By assuming that noise in the pairwise distance measurements scale linearly with\ndistance, both the mean and variance of the Gaussian describing this distribution is ef\ufb01ciently encoded\nin a single PC-PC synapse.\n\n(6)\n\nij\n\nOf\ufb02ine message passing for probabilistic structural inference The map con\ufb01guration in Eq. 5\nis approximated by message passing between PCs via the belief propagation algorithm (BP) [47], a\nsingle update cycle consisting of a message broadcast and a belief update. A message is de\ufb01ned as\nthe probability distribution of a receiving node given the broadcasting node\u2019s belief and the pairwise\npotential \u03c8ij between the two (see below). Firstly, at iteration n the node t integrates all messages\nu\u2192t(bt) received from its neighbours u \u2208 \u0393t with its prior self-belief B(0)\n(bt) to compute its\nm(n)\nupdated self-belief:\n\nt\n\nB(n)\n\nt\n\n(bt) \u221d B(0)\n\nt\n\n(bt)\n\nm(n)\n\nu\u2192t(bt)\n\n(7)\n\n(cid:89)\n\nu\u2208\u0393t\n\nEq. 7 therefore represents the belief of node t over its own state (location) given all messages from\nits connected neighbours in the graph and its prior. Secondly, node t broadcasts messages back to its\nneighbours expressing its belief over their states:\n\nt\u2192u (bu) \u221d\nm(n+1)\n\n\u03c8tu(bt, bu)B(n)\n\nt\n\n(bt)/m(n)\n\nu\u2192t(bt)dbt\n\n(8)\n\n(cid:90)\n\nwhere the new message is divided by the reciprocal message from the previous iteration [47].\n\n5\n\n\fFigure 3: The loop-closure\ntask. A The agent navigates a\nnovel circular track, accumu-\nlating PI error. Lap comple-\ntion (iii) triggers an of\ufb02ine in-\nference event (see main text\nand Supp. Video 1) for de-\ntails). B Structure inferred af-\nter loop-closure. C PE is re-\nduced on completion of sub-\nsequent laps. D Of\ufb02ine infer-\nence allows one-shot learn-\ning when compared to the on-\nline system.\n\nPrincipled message scheduling In a naiive \u2018sequential\u2019 schedule, all PCs broadcast messages\nbefore updating their beliefs. Instead, we implement an asynchronous message schedule (\u2018Max-\nEntropy\u2019) in which only cells whose belief has changed by some threshold amount broadcast\nmessages at the next time-step [11]. The \u2018message tension\u2019 T n\nis de\ufb01ned by the cumulative Jensen-\ni\nShannon divergence (symmetric KL; see Appendix E) between beliefs at successive time-steps:\ni = T n\u22121\n). When the message tension is below a prede\ufb01ned threshold Tmin, a\nT n\nnode is considered converged and stops broadcasting messages. A single of\ufb02ine inference event is\nde\ufb01ned by the convergence of all nodes of the graph.\n\ni ||bn\u22121\n\ni\n\n+ JS(bn\n\ni\n\n3.4 Prediction errors as an arbitration mechanism\n\nRather than continually perform map updates, we propose a more computationally (and energetically)\nfavourable scheme in which the of\ufb02ine system is only recruited when the online system is performing\npoorly (batch updates are also known to be more robust [1]). We de\ufb01ne the \u2018prediction error\u2019 (PE) of\nt) \u2212 H(ptB), to compare the predicted and observed GC distribution,\nthe online system: Et = H(G(cid:48)\nwhere H(\u00b7) is the information entropy such that the PE term is positive when the inbound sensory\ninformation has a lower entropy than the current location estimate. Of\ufb02ine inference events are then\ninitiated by positive PEs above a threshold E 0. Note that the form of the PE update rule is similar\nbut not identical to the rule for broadcasting messages during of\ufb02ine inference; sensory input that\nincreases the entropy should not trigger of\ufb02ine inference events.\n\n4 Results\n\n4.1\n\nInference on static structures\n\nWe \ufb01rst tested the ability of the of\ufb02ine system to infer the structure of three environments (Fig. 2).\nGiven erroneous initial estimates corresponding to priors formed during noisy PI, the system is able\nto correctly infer the true structures as those that satis\ufb01ed pairwise measurements between states\n(Fig. 2D). However, an immediate consequence of the system is that this inferred structure will be\nsensitive to topology. Although PI will impose metric priors, where these priors are unreliable (as in\nthe case of navigating around an unfamiliar ring environment under noisy PI), the inferred structure\nis sensitive to the \u2018closure\u2019 of loops (Fig. 2D, \"Broken Ring\").\n\n4.2 Loop closure experiment\n\nIn the loop closure task (Fig. 3; Supp. Video 1), place \ufb01elds are distributed uniformly around a circular\n1D track. Initial location con\ufb01dence is high, such that place and GCs active at the start location\n(0 rads) form strong associations. As the agent navigates around the track, PI error accumulates and\nthe con\ufb01dence in location decreases, resulting in subsequent PC-GC associations becoming more\ndiffuse and less likely to correspond to the true structure (Fig. 3Ai). Due to the accumulated error,\nwhen the agent completes a full lap it receives a sharp input from the PCs initially active at the starting\nlocation, producing a strong positive PE and triggering an of\ufb02ine inference event (Fig. 3Aiii).\n\n6\n\n\fFigure 4: A Principled message scheduling generates PC sequences. Nodes are connected via their\npairwise potentials \u03c8. (t=0) Sensory input causes an update to the belief of node A. (t=1) A sends\na message to B causing it to update its belief. (t=2) Messages from B only cause C to update its\nbelief, so only C broadcasts at the next time-step. B The \u2018Max-Entropy\u2019 schedule converges faster\nthan when all PCs broadcast messages at each time-step. C Examples of PC reactivation sequences.\nMultiple sequences occur simultaneously (Left) and become longer and smoother when pairwise\nmeasurements are less con\ufb01dent (Right; E, F). D Forward and reverse sequences occurred equally.\n\nOf\ufb02ine inference allows one-shot learning As expected, structural error is reduced signi\ufb01cantly\nfollowing the triggered of\ufb02ine inference events. This reduction is markedly larger than in equivalent\ntrials using only the online system, resembling a \u2018one-shot\u2019 learning process (Fig 3D). Given the\nrapid map-learning, PEs on subsequent laps are also reduced (Fig. 3C).\n\nPrincipled message scheduling produces structured reactivations BP seeks a solution whereby\nmessages received from neighbouring nodes cause negligible change to the receiving nodes\u2019 belief.\nThe scheduling is therefore important from an energetic perspective; messages that do not produce\nchanges in the beliefs of neighbours are redundant. Nodes which did not signi\ufb01cantly update their\nself-beliefs following receipt of a message therefore do not need to re-broadcast a message at the\nnext time-step (Fig. 4A).\nIn addition to the energetic advantages, the \u2018Max-Entropy\u2019 schedule also contributes to inference\nperformance, converging faster than a simple \u2018sequential\u2019 scheme in which all nodes broadcast\nmessages at each time-step, despite broadcasting fewer total messages (Fig. 4B).\nThe sequences of reactivations also contained signi\ufb01cant structure, tending to propagate initially\nbackwards along the track from the animal\u2019s current position, resembling the PC reactivations\nduring reverse hippocampal replay (Fig. 4C) [15]. Sequences did not always hop to adjacent\n\ufb01elds, occasionally hopping to new locations where remote sequences were then initiated (Fig. 4C)\n[8; 27; 53]. Multiple sequences at different remote locations can be seen to occur simultaneously or\nin an alternating fashion (Fig. 4C) [27]. Both forward and reverse sequences were observed in equal\nproportion (Fig. 4D) [15; 9]. Lastly, the \u2018hoppiness\u2019 of the sequences was related to the con\ufb01dence in\nthe pairwise observations, information propagating more quickly and smoothly in a \u2018stiffer\u2019 graph (a\ngraph with more con\ufb01dent pairwise observations; Fig. 4E,F) [49; 27; 53].\n\n4.3 Local distortions to the cognitive map\n\nGrid patterns undergo signi\ufb01cant local distortions in open environments, decreasing in scale and be-\ncoming less uniform (more sheared) towards the corners [23]. We hypothesized that these distortions\nmight re\ufb02ect the underlying structure of the environment as captured in the associative structure in\nCA3 and manifested in its projections to metric GC space.\nIn the same study, scale was also positively correlated to behavioural occupancy (animals spent more\ntime in the middle of the environment; Fig. 5B, E bottom row, Appendix Fig. 2B) [23]. This effect\nwas mirrored in our model, since over-sampling of the tails of the place \ufb01elds near the boundaries\nof the environment led to the associated PCs overestimating their pairwise distances (Appendix Fig.\n\n7\n\n\fFigure 5: Distortions to the\ncognitive map. A Variation\nin place \ufb01eld shape results\nin distortions in the GC \ufb01r-\ning pattern (D,E, Top). B\nLearned distances due to bi-\nased sampling of the environ-\nment [23] also produce local\ndistortions (D,E, Bottom). C\nInferred structure in CA3.\n\nFigure 6: A Neural model of coordinated HPC-mEC replay. A (1D) The broadcasting PC PB sends\na spike to neighbour PR (CA3), at the same time initiating a travelling wave in the GCs (mEC) by\nvirtue of its synaptic projections B. (Left) No learning occurs when the spike and travelling wave\narrive at PR at the same time. (Right) If the CA3 spike arrives ahead of the travelling wave, the\nsynaptic associations of PR are adjusted towards the currently active GCs. B Comparison of the\n\u2018algorithmic\u2019 [32] and neural BP implementations. C Travelling waves on the 2D GC sheet.\n\n2B); their mean co-\ufb01ring was lower than expected if the animal were to sample from the place \ufb01eld\nuniformly; Fig. 2A). Note that pairwise associative distance is inversely related to the scale of the\ngrid pattern readout, since larger associative distance implies travelling further in metric space (see\nAppendix D.1 and Appendix Fig. 2C)\nGiven that PC \ufb01ring is related to the con\ufb01dence of the presence of speci\ufb01c sensory cues, we also\nexplored the case where place \ufb01elds were sharper near the edges of an environment, as would be\nthe case if driven by strong geometric cues (Fig. 5A, top row) [6; 25; 24]. These non-uniformities\nalso produced the same local warping of the grid pattern Fig. 5C, D, E top row). This latter effect\nis attributed to the nature of Hebbian learning rules, whose learned synaptic strengths re\ufb02ect the\nvariance normalized distance between the \ufb01elds, as opposed to the true Euclidean distance (Fig. 2A;\nAppendix D.1) [4]. Our model suggests that the cognitive \u2018distance\u2019 (or \u2018discriminability\u2019) between\ntwo sensory stimuli should be greater if the absolute con\ufb01dence in the locations of each is greater.\n\n4.4 A neural-level model of coordinated place-GC replay\n\nHow might belief propagation be implemented in the brain? More speci\ufb01cally, how might \u2018message\nbroadcasts\u2019 correspond to spikes \ufb01red by PCs during replay, and how would GCs contribute to of\ufb02ine\ninference? Our proposed mechanism relies on coincidence detection by a \u2018receiving\u2019 PC PR of a\ndirect spike from a \u2018broadcasting\u2019 PC PB and a travelling wave of activity across the GC population\n(Fig. 6A,C and Supp. Video 2).\nA message broadcast is initiated by the \ufb01ring of a spike from PB to synaptically connected PCs,\nwith a transmission delay proportional to the inferred pairwise distance dij (Fig. 6A, \"Place cells\").\nIn parallel, the same spike from PB drives activity in the GC population via the PC-GC synaptic\nassociations (Fig. 6A, \"Grid cells\"). This activity propagates radially outwards at a constant speed,\nidentically to PI during online\naccumulating noise in proportion to the distance travelled (i.e.\nlocalization; Fig. 6C; see Appendix D.2). This can be viewed as activity propagating through two\ngenerative models of associative and metric space (see Appendix D.3).\n\n8\n\n\fWhen PR receives the spike from PB, we assume that the depolarization causes learning between PR\nand the currently active GCs, even though PR does not necessarily \ufb01re a spike [20; 18; 59]. If the\ndistance indicated by the relative propagation of activity between GCs corresponding to the synaptic\nprojections of PB and PR is equal to the distance encoded by the recurrent association between PB\nand PR, PR will receive the spike from PB at the same time that the travelling wave arrives at the\nGCs to which PR projects, so that no signi\ufb01cant synaptic changes are produced (Fig. 6A, Left). If\nthese two distances are in disagreement, PR will revise its belief, shifting its synaptic associations to\n\u2018earlier\u2019 or \u2018later\u2019 GCs, respectively (Fig. 6C, Right).\nLastly, \ufb01ring of PR is triggered only if there is signi\ufb01cant change in its synaptic weights to the GC\npopulation, i.e. only messages indicating belief changes are propagated (a similar condition to that\nused to initiate the of\ufb02ine system). We propose therefore that the \u2018message tension\u2019 term, which\ngoverns spiking, might correspond to the accumulation of a learning related neuromodulator.\n\n5 Discussion\n\nDuring active exploration, place and grid cells are predominantly active when the location of the\nanimal corresponds to their spatial receptive \ufb01elds. During periods of rest or immobility however,\nthe same cells have nonetheless been observed to reactivate at remote locations. That these distinct\nregimes of neural activity correspond also to distinct behavioural states suggests a functional role for\nonline and of\ufb02ine processing in the HPC-mEC system.\nOurs is the \ufb01rst model to demonstrate how PI and sensory based estimates could interact probabilisti-\ncally during online localization in the HPC-mEC system (Fig. 4). We also show how more complex\nprobabilistic inference could be performed via the of\ufb02ine interaction of HPC and mEC (Fig. 2, 6)\nand propose a detailed mapping of the joint map-location distribution to physiological correlates\n(Appendix Fig. 1). We then show how prediction errors between predicted and observed sensory\nstimuli can be used to ef\ufb01ciently arbitrate between the two systems (Fig. 3).\nOf\ufb02ine inference events based on principled message passing resemble PC reactivations during replay\nevents, which occur during of\ufb02ine behaviours such as pausing or sleep [15]. Our model predicts\ntherefore that replay events (and associated sharp wave ripples) should be more frequent during\nstructural changes to the environment rather than being solely responsive to reward [53], although\nrewards themselves may constitute salient sensory stimuli (an apple is highly indicative of location in\nan otherwise featureless maze). Our algorithmic and neural models [32] of this process are the \ufb01rst\nto predict the detailed interaction between PCs and GCs during coordinated replay events [45; 64].\nAlthough investigated in a spatial context, structured information propagation may be a general\nmechanism for embedding associative experience in metric space [28; 19].\nOur model is also the \ufb01rst to propose that observed local distortions to the grid pattern [23], re\ufb02ect\nthe underlying associative structure of the environment. Place \ufb01elds are known to be smaller and\nmore dense near to boundaries and salient locations [26]. Warping of the grid scale would thus be\nconsistent with preserving a constant rate of change of sensory information [63]. The HPC-mEC\ninteraction can be interpreted therefore as the embedding of associative structure within a metric map,\nto allow the agent to determine shortcuts between previously unexperienced state transitions [51].\n\nAcknowledgements\n\nWe acknowledge funding from European Union\u2019s Horizon 2020 research and innovation programme\nHuman Brain Project SGA2 (grant agreement no. 785907), Wellcome and ERC Advanced grant\nNEUROMEM.\n\nReferences\n[1] Tim Bailey and Hugh Durrant-Whyte. Simultaneous localization and mapping (slam): Part ii.\n\nIEEE Robotics & Automation Magazine, 13(3):108\u2013117, 2006.\n\n[2] Andrea Banino, Caswell Barry, Benigno Uria, Charles Blundell, Timothy Lillicrap, Piotr\nMirowski, Alexander Pritzel, Martin J Chadwick, Thomas Degris, Joseph Modayil, et al. Vector-\nbased navigation using grid-like representations in arti\ufb01cial agents. 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