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Summary: How the topography of neural circuits relates to their function remains unclear. Although topographic maps exist for sensory and motor variables, they are rarely observed for cognitive variables. Using calcium imaging during virtual navigation, we investigated the relationship between the anatomical organization and functional properties of grid cells, which represent a cognitive code for location during navigation. We found a substantial degree of grid cell micro-organization in mouse medial entorhinal cortex: grid cells and modules all clustered anatomically. Within a module, the layout of grid cells was a noisy two-dimensional lattice, in which the anatomical distribution of grid cells largely matched their spatial tuning phases. This micro-arrangement of phases demonstrates the existence of a topographical map encoding a cognitive variable in rodents. It contributes to a foundation for evaluating circuit models of the grid cell network, and is consistent with continuous attractor models as the mechanism of grid formation.

SUMMARY Grid cells, defined by their striking periodic spatial responses in open 2D arenas, appear to respond differently on 1D tracks: the multiple response fields are not periodically arranged, peak amplitudes vary across fields, and the mean spacing between fields is larger than in 2D environments. We ask whether such 1D responses are consistent with the system’s 2D dynamics. Combining analytical and numerical methods, we show that the 1D responses of grid cells with stable 1D fields are consistent with a linear slice through a 2D triangular lattice. Further, the 1D responses of comodular cells are well described by parallel slices, and the offsets in the starting points of the 1D slices can predict the measured 2D relative spatial phase between the cells. From these results, we conclude that the 2D dynamics of these cells is preserved in 1D, suggesting a common computation during both types of navigation behavior.

Hippocampal neurons fire selectively in local behavioral contexts such as the position in an environment or phase of a task, 1-3 and are thought to form a cognitive map of task-relevant variables. 1, 4, 5 However, their activity varies over repeated behavioral conditions, 6 such as di↵erent runs through the same position or repeated trials. Although widely observed across the brain, 7-10 such variability is not well understood, and could reflect noise or structure, such as the encoding of additional cognitive information. 6,[11][12][13] Here, we introduce a conceptual model to explain variability in terms of underlying, population-level structure in single-trial neural activity. To test this model, we developed a novel unsupervised learning algorithm incorporating temporal dynamics, in order to characterize population activity as a trajectory on a nonlinear manifold-a space of possible network states. The manifold's structure captures correlations between neurons and temporal relationships between states, constraints arising from underlying network architecture and inputs. Using measurements of activity over time but no information about exogenous behavioral variables, we recovered hippocampal activity manifolds during spatial and non-spatial cognitive tasks in rats. Manifolds were low-dimensional and smoothly encoded task-related variables, but contained an extra dimension reflecting information beyond the measured behavioral variables. Consistent with our model, neurons fired as a function of overall network state, and fluctuations in their activity across trials corresponded to variation in the underlying trajectory on the manifold. In particular, the extra dimension allowed the system to take di↵erent trajectories despite repeated behavioral conditions. Furthermore, the trajectory could temporarily decouple from current behavioral conditions and traverse neighboring manifold points corresponding to past, future, or nearby behavioral states. Our results suggest that trial-to-trial variability in the hippocampus is structured, and may reflect the operation of internal cognitive processes. The manifold structure of population activity is well-suited for organizing information to support memory, 1, 5, 14 planning, 12, 15, 16 and reinforcement learning. 17, 18 In general, our approach could find broader use in probing the organization and computational role of circuit dynamics in other brain regions.

We consider a threshold-crossing spiking process as a simple model for the activity within a population of neurons. Assuming that these neurons are driven by a common fluctuating input with gaussian statistics, we evaluate the cross-correlation of spike trains in pairs of model neurons with different thresholds. This correlation function tends to be asymmetric in time, indicating a preference for the neuron with the lower threshold to fire before the one with the higher threshold, even if their inputs are identical. The relationship between these results and spike statistics in other models of neural activity is explored. In particular, we compare our model with an integrate-and-fire model in which the membrane voltage resets following each spike. The qualitative properties of spike cross-correlations, emerging from the threshold-crossing model, are similar to those of bursting events in the integrate-and-fire model. This is particularly true for generalized integrate-and-fire models in which spikes tend to occur in bursts, as observed, for example, in retinal ganglion cells driven by a rapidly fluctuating visual stimulus. The threshold-crossing model thus provides a simple, analytically tractable description of event onsets in these neurons.

We characterize the (1, 1) knots in the three-sphere and lens spaces that admit non-trivial L-space surgeries. As a corollary, 1-bridge braids in these manifolds admit nontrivial L-space surgeries. We also recover a characterization of the Berge manifold amongst 1-bridge braid exteriors. Introduction.An L-space is a rational homology sphere Y with the "simplest" Heegaard Floer invariant: HF (Y ) is a free abelian group of rank |H 1 (Y ; Z)|. Examples abound and include lens spaces and, more generally, connected sums of manifolds with elliptic geometry [OS05]. One of the most prominent problems in relating Heegaard Floer homology to low-dimensional topology is to give a topological characterization of L-spaces. Work by many researchers has synthesized a bold and intriguing proposal that seeks to do so in terms of taut foliations and orderability of the fundamental group [Juh15, Conjecture 5].A prominent source of L-spaces arises from surgeries along knots. Suppose that K is a knot in a closed three-manifold Y . If K admits a non-trivial surgery to an L-space, then K is an L-space knot. Examples include torus knots and, more generally, Berge knots in S 3 [Ber]; two more constructions especially pertinent to our work appear in [HLV14,Vaf15]. If an L-space knot K admits more than one L-space surgery -for instance, if Y itself is an L-space -then it admits an interval of L-space surgery slopes, so it generates abundant examples of L-spaces [RR15]. With the lack of a compelling guiding conjecture as to which knots are L-space knots, and as a probe of the L-space conjecture mentioned above, it is valuable to catalog which knots in various special families are L-space knots. This is the theme of the present work.The manifolds in which we operate are the rational homology spheres that admit a genus one Heegaard splitting, namely the three-sphere and lens spaces. The knots we consider are the (1, 1) knots in these spaces: these are the knots that can be isotoped to meet each Heegaard solid torus in a properly embedded, boundary-parallel arc. Our main result, Theorem 1.2 below, characterizes (1, 1) L-space knots in simple, diagrammatic terms.A (1, 1) diagram is a doubly-pointed Heegaard diagram (Σ, α, β, z, w), where (Σ, α, β) is a genus one Heegaard diagram of a 3-manifold Y . The (1, 1) knots in Y are precisely those that admit a doubly-pointed Heegaard diagram [GMM05, Hed11, Ras05]. A (1, 1) diagram is reduced if every bigon contains a basepoint. We can transform a given (1, 1) diagram of K into a reduced (1, 1) diagram of K by isotoping the curves into minimal position in the complement of the basepoints: we accomplish this by successively isotoping away bigons in 1 arXiv:1610.04810v2 [math.GT]

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