Reconceiving the hippocampal map as a topological template

Computational role of structure in neural activity and connectivity

One major challenge of neuroscience is identifying structure in seemingly disorganized neural activity. Different types of structure have different computational implications that can help neuroscientists understand the functional role of a particular brain area. Here, we outline a unified approach to characterize structure by inspecting the representational geometry and the modularity properties of the recorded activity and show that a similar approach can also reveal structure in connectivity. We start by setting up a general framework for determining geometry and modularity in activity and connectivity and relating these properties with computations performed by the network. We then use this framework to review the types of structure found in recent studies of model networks performing three classes of computations.

Topological structure of population activity in mouse visual cortex encodes densely sampled stimulus rotations

The primary visual cortex is one of the most well understood regions supporting the processing involved in sensory computation. Following the popularization of high-density neural recordings, it has been observed that the activity of large neural populations is often constrained to low dimensional manifolds. In this work, we quantify the structure of such neural manifolds in the visual cortex. We do this by analyzing publicly available two-photon optical recordings of mouse primary visual cortex in response to visual stimuli with a densely sampled rotation angle. Using a geodesic metric along with persistent homology, we discover that population activity in response to such stimuli generates a circular manifold, encoding the angle of rotation. Furthermore, we observe that this circular manifold is expressed differently in subpopulations of neurons with differing orientation and direction selectivity. Finally, we discuss some of the obstacles to reliably retrieving the truthful topology generated by a neural population.

Neural Trajectories of Conceptually Related Events

In a series of conceptually related episodes, meaning arises from the link between these events rather than from each event individually. How does the brain keep track of conceptually related sequences of events (i.e., conceptual trajectories)? In a particular kind of conceptual trajectory-a social relationship-meaning arises from a specific sequence of interactions. To test whether such abstract sequences are neurally tracked, we had participants complete a naturalistic narrative-based social interaction game, during functional magnetic resonance imaging. We modeled the simulated relationships as trajectories through an abstract affiliation and power space. In two independent samples, we found evidence of individual social relationships being tracked with unique sequences of hippocampal states. The neural states corresponded to the accumulated trial-to-trial affiliation and power relations between the participant and each character, such that each relationship's history was captured by its own neural trajectory. Each relationship had its own sequence of states, and all relationships were embedded within the same manifold. As such, we show that the hippocampus represents social relationships with ordered sequences of low-dimensional neural patterns. The number of distinct clusters of states on this manifold is also related to social function, as measured by the size of real-world social networks. These results suggest that our evolving relationships with others are represented in trajectory-like neural patterns.

Efficient, continual, and generalized learning in the brain - neural mechanism of Mental Schema 2.0

There has been tremendous progress in artificial neural networks (ANNs) over the past decade; however, the gap between ANNs and the biological brain as a learning device remains large. With the goal of closing this gap, this paper reviews learning mechanisms in the brain by focusing on three important issues in ANN research: efficiency, continuity, and generalization. We first discuss the method by which the brain utilizes a variety of self-organizing mechanisms to maximize learning efficiency, with a focus on the role of spontaneous activity of the brain in shaping synaptic connections to facilitate spatiotemporal learning and numerical processing. Then, we examined the neuronal mechanisms that enable lifelong continual learning, with a focus on memory replay during sleep and its implementation in brain-inspired ANNs. Finally, we explored the method by which the brain generalizes learned knowledge in new situations, particularly from the mathematical generalization perspective of topology. Besides a systematic comparison in learning mechanisms between the brain and ANNs, we propose "Mental Schema 2.0," a new computational property underlying the brain's unique learning ability that can be implemented in ANNs.

Metric information in cognitive maps: Euclidean embedding of non-Euclidean environments

The structure of the internal representation of surrounding space, the so-called cognitive map, has long been debated. A Euclidean metric map is the most straight-forward hypothesis, but human navigation has been shown to systematically deviate from the Euclidean ground truth. Vector navigation based on non-metric models can better explain the observed behavior, but also discards useful geometric properties such as fast shortcut estimation and cue integration. Here, we propose another alternative, a Euclidean metric map that is systematically distorted to account for the observed behavior. The map is found by embedding the non-metric model, a labeled graph, into 2D Euclidean coordinates. We compared these two models using data from a human behavioral study where participants had to learn and navigate a non-Euclidean maze (i.e., with wormholes) and perform direct shortcuts between different locations. Even though the Euclidean embedding cannot correctly represent the non-Euclidean environment, both models predicted the data equally well. We argue that the embedding naturally arises from integrating the local position information into a metric framework, which makes the model more powerful and robust than the non-metric alternative. It may therefore be a better model for the human cognitive map.

A manifold neural population code for space in hippocampal coactivity dynamics independent of place fields

Hippocampus place cell discharge is temporally unreliable across seconds and days, and place fields are multimodal, suggesting an "ensemble cofiring" spatial coding hypothesis with manifold dynamics that does not require reliable spatial tuning, in contrast to hypotheses based on place field (spatial tuning) stability. We imaged mouse CA1 (cornu ammonis 1) ensembles in two environments across three weeks to evaluate these coding hypotheses. While place fields "remap," being more distinct between than within environments, coactivity relationships generally change less. Decoding location and environment from 1-s ensemble location-specific activity is effective and improves with experience. Decoding environment from cell-pair coactivity relationships is also effective and improves with experience, even after removing place tuning. Discriminating environments from 1-s ensemble coactivity relies crucially on the cells with the most anti-coactive cell-pair relationships because activity is internally organized on a low-dimensional manifold of non-linear coactivity relationships that intermittently reregisters to environments according to the anti-cofiring subpopulation activity.

How our understanding of memory replay evolves

Memory reactivations and replay, widely reported in the hippocampus and cortex across species, have been implicated in memory consolidation, planning, and spatial and skill learning. Technological advances in electrophysiology, calcium imaging, and human neuroimaging techniques have enabled neuroscientists to measure large-scale neural activity with increasing spatiotemporal resolution and have provided opportunities for developing robust analytic methods to identify memory replay. In this article, we first review a large body of historically important and representative memory replay studies from the animal and human literature. We then discuss our current understanding of memory replay functions in learning, planning, and memory consolidation and further discuss the progress in computational modeling that has contributed to these improvements. Next, we review past and present analytic methods for replay analyses and discuss their limitations and challenges. Finally, looking ahead, we discuss some promising analytic methods for detecting nonstereotypical, behaviorally nondecodable structures from large-scale neural recordings. We argue that seamless integration of multisite recordings, real-time replay decoding, and closed-loop manipulation experiments will be essential for delineating the role of memory replay in a wide range of cognitive and motor functions.

Rethinking retrosplenial cortex: Perspectives and predictions

The last decade has produced exciting new ideas about retrosplenial cortex (RSC) and its role in integrating diverse inputs. Here, we review the diversity in forms of spatial and directional tuning of RSC activity, temporal organization of RSC activity, and features of RSC interconnectivity with other brain structures. We find that RSC anatomy and dynamics are more consistent with roles in multiple sensorimotor and cognitive processes than with any isolated function. However, two more generalized categories of function may best characterize roles for RSC in complex cognitive processes: (1) shifting and relating perspectives for spatial cognition and (2) prediction and error correction for current sensory states with internal representations of the environment. Both functions likely take advantage of RSC's capacity to encode conjunctions among sensory, motor, and spatial mapping information streams. Together, these functions provide the scaffold for intelligent actions, such as navigation, perspective taking, interaction with others, and error detection.

Unusual Mathematical Approaches Untangle Nervous Dynamics

The massive amount of available neurodata suggests the existence of a mathematical backbone underlying neuronal oscillatory activities. For example, geometric constraints are powerful enough to define cellular distribution and drive the embryonal development of the central nervous system. We aim to elucidate whether underrated notions from geometry, topology, group theory and category theory can assess neuronal issues and provide experimentally testable hypotheses. The Monge’s theorem might contribute to our visual ability of depth perception and the brain connectome can be tackled in terms of tunnelling nanotubes. The multisynaptic ascending fibers connecting the peripheral receptors to the neocortical areas can be assessed in terms of knot theory/braid groups. Presheaves from category theory permit the tackling of nervous phase spaces in terms of the theory of infinity categories, highlighting an approach based on equivalence rather than equality. Further, the physical concepts of soft-matter polymers and nematic colloids might shed new light on neurulation in mammalian embryos. Hidden, unexpected multidisciplinary relationships can be found when mathematics copes with neural phenomena, leading to novel answers for everlasting neuroscientific questions. For instance, our framework leads to the conjecture that the development of the nervous system might be correlated with the occurrence of local thermal changes in embryo–fetal tissues.

The hippocampal formation as a hierarchical generative model supporting generative replay and continual learning

We advance a novel computational theory of the hippocampal formation as a hierarchical generative model that organizes sequential experiences, such as rodent trajectories during spatial navigation, into coherent spatiotemporal contexts. We propose that the hippocampal generative model is endowed with inductive biases to identify individual items of experience (first hierarchical layer), organize them into sequences (second layer) and cluster them into maps (third layer). This theory entails a novel characterization of hippocampal reactivations as generative replay: the offline resampling of fictive sequences from the generative model, which supports the continual learning of multiple sequential experiences. We show that the model learns and efficiently retains multiple spatial navigation trajectories, by organizing them into spatial maps. Furthermore, the model reproduces flexible and prospective aspects of hippocampal dynamics that are challenging to explain within existing frameworks. This theory reconciles multiple roles of the hippocampal formation in map-based navigation, episodic memory and imagination.

Adaptive cognitive maps for curved surfaces in the 3D world

Terrains in a 3D world can be undulating. Yet, most prior research has exclusively investigated spatial representations on a flat surface, leaving a 2D cognitive map as the dominant model in the field. Here, we investigated whether humans represent a curved surface by building a dimension-reduced flattened 2D map or a full 3D map. Participants learned the location of objects positioned on a flat and curved surface in a virtual environment by driving on the concave side of the surface (Experiment 1), driving and looking vertically (Experiment 2), or flying (Experiment 3). Subsequently, they were asked to retrieve either the path distance or the 3D Euclidean distance between the objects. Path distance estimation was good overall, but we found a significant underestimation bias for the path distance on the curve, suggesting an influence of potential 3D shortcuts, even though participants were only driving on the surface. Euclidean distance estimation was better when participants were exposed more to the global 3D structure of the environment by looking and flying. These results suggest that the representation of the 2D manifold, embedded in a 3D world, is neither purely 2D nor 3D. Rather, it is flexible and dependent on the behavioral experience and demand.

Spatial representability of neuronal activity

A common approach to interpreting spiking activity is based on identifying the firing fields—regions in physical or configuration spaces that elicit responses of neurons. Common examples include hippocampal place cells that fire at preferred locations in the navigated environment, head direction cells that fire at preferred orientations of the animal’s head, view cells that respond to preferred spots in the visual field, etc. In all these cases, firing fields were discovered empirically, by trial and error. We argue that the existence and a number of properties of the firing fields can be established theoretically, through topological analyses of the neuronal spiking activity. In particular, we use Leray criterion powered by persistent homology theory, Eckhoff conditions and Region Connection Calculus to verify consistency of neuronal responses with a single coherent representation of space.

Neural population geometry: An approach for understanding biological and artificial neural networks

Advances in experimental neuroscience have transformed our ability to explore the structure and function of neural circuits. At the same time, advances in machine learning have unleashed the remarkable computational power of artificial neural networks (ANNs). While these two fields have different tools and applications, they present a similar challenge: namely, understanding how information is embedded and processed through high-dimensional representations to solve complex tasks. One approach to addressing this challenge is to utilize mathematical and computational tools to analyze the geometry of these high-dimensional representations, i.e., neural population geometry. We review examples of geometrical approaches providing insight into the function of biological and artificial neural networks: representation untangling in perception, a geometric theory of classification capacity, disentanglement, and abstraction in cognitive systems, topological representations underlying cognitive maps, dynamic untangling in motor systems, and a dynamical approach to cognition. Together, these findings illustrate an exciting trend at the intersection of machine learning, neuroscience, and geometry, in which neural population geometry provides a useful population-level mechanistic descriptor underlying task implementation. Importantly, geometric descriptions are applicable across sensory modalities, brain regions, network architectures, and timescales. Thus, neural population geometry has the potential to unify our understanding of structure and function in biological and artificial neural networks, bridging the gap between single neurons, population activities, and behavior.

Graded remapping of hippocampal ensembles under sensory conflicts

Hippocampal place cells are thought to constitute a cognitive map of space derived from multimodal sensory inputs. Alteration of allocentric (visual) cues in a fixed environment is known to induce modulations of place cell activity to varying degrees from rate changes to global remapping. To determine how hippocampal ensembles combine multimodal sensory cues, we examine hippocampal CA1 remapping in Mongolian gerbils in a 1D virtual reality experiment, during which self-motion cues (locomotor, vestibular, and optic flow information) and allocentric visual cues are altered. We observe that self-motion cues are over-represented, but responsiveness to allocentric visual cues, although task-irrelevant, elicits both rate and global remapping in the hippocampal ensemble. We propose that remapping can be reconciled by considering global, partial, and rate remapping on a continuous scale on which the graded change of activity in the entire CA1 population can be interpreted as the expectancy about the animal's spatial environment.

The grid code for ordered experience

Entorhinal cortical grid cells fire in a periodic pattern that tiles space, which is suggestive of a spatial coordinate system. However, irregularities in the grid pattern as well as responses of grid cells in contexts other than spatial navigation have presented a challenge to existing models of entorhinal function. In this Perspective, we propose that hippocampal input provides a key informative drive to the grid network in both spatial and non-spatial circumstances, particularly around salient events. We build on previous models in which neural activity propagates through the entorhinal-hippocampal network in time. This temporal contiguity in network activity points to temporal order as a necessary characteristic of representations generated by the hippocampal formation. We advocate that interactions in the entorhinal-hippocampal loop build a topological representation that is rooted in the temporal order of experience. In this way, the structure of grid cell firing supports a learned topology rather than a rigid coordinate frame that is bound to measurements of the physical world.

Simplicial and topological descriptions of human brain dynamics

While brain imaging tools like functional magnetic resonance imaging (fMRI) afford measurements of whole-brain activity, it remains unclear how best to interpret patterns found amid the data’s apparent self-organization. To clarify how patterns of brain activity support brain function, one might identify metric spaces that optimally distinguish brain states across experimentally defined conditions. Therefore, the present study considers the relative capacities of several metric spaces to disambiguate experimentally defined brain states. One fundamental metric space interprets fMRI data topographically, that is, as the vector of amplitudes of a multivariate signal, changing with time. Another perspective compares the brain’s functional connectivity, that is, the similarity matrix computed between signals from different brain regions. More recently, metric spaces that consider the data’s topology have become available. Such methods treat data as a sample drawn from an abstract geometric object. To recover the structure of that object, topological data analysis detects features that are invariant under continuous deformations (such as coordinate rotation and nodal misalignment). Moreover, the methods explicitly consider features that persist across multiple geometric scales. While, certainly, there are strengths and weaknesses of each brain dynamics metric space, wefind that those that track topological features optimally distinguish experimentally defined brain states.

Time-varying functional connectivity interprets brain function as time-varying patterns of coordinated brain activity. While many questions remain regarding how brain function emerges from multiregional interactions, advances in the mathematics of topological data analysis (TDA) may provide new insights. One tool from TDA, “persistent homology,” observes the occurrence and persistence of n-dimensional holes in a sequence of simplicial complexes extracted from a weighted graph. In the present study, we compare the use of persistent homology versus more traditional metrics at the task of segmenting brain states that differ across experimental conditions. We find that the structures identified by persistent homology more accurately segment the stimuli, more accurately segment high versus low performance levels under common stimuli, and generalize better across volunteers.

Hippocampal place cells encode global location but not connectivity in a complex space

Flexible navigation relies on a cognitive map of space, thought to be implemented by hippocampal place cells: neurons that exhibit location-specific firing. In connected environments, optimal navigation requires keeping track of one's location and of the available connections between subspaces. We examined whether the dorsal CA1 place cells of rats encode environmental connectivity in four geometrically identical boxes arranged in a square. Rats moved between boxes by pushing saloon-type doors that could be locked in one or both directions. Although rats demonstrated knowledge of environmental connectivity, their place cells did not respond to connectivity changes, nor did they represent doorways differently from other locations. Place cells coded location in a global reference frame, with a different map for each box and minimal repetitive fields despite the repetitive geometry. These results suggest that CA1 place cells provide a spatial map that does not explicitly include connectivity.

Cognitive Maps: Distortions of the Hippocampal Space Map Define Neighborhoods

In place of continuous overhead satellite views of an environment, the brain often relies on first-person experiences to estimate spatial relationships between locations. Using new methods, a recent study has found the spatial metric observed in hippocampal activity adapts to encode local environmental terrain.

From Topological Analyses to Functional Modeling: The Case of Hippocampus

Topological data analyses are widely used for describing and conceptualizing large volumes of neurobiological data, e.g., for quantifying spiking outputs of large neuronal ensembles and thus understanding the functions of the corresponding networks. Below we discuss an approach in which convergent topological analyses produce insights into how information may be processed in mammalian hippocampus—a brain part that plays a key role in learning and memory. The resulting functional model provides a unifying framework for integrating spiking data at different timescales and following the course of spatial learning at different levels of spatiotemporal granularity. This approach allows accounting for contributions from various physiological phenomena into spatial cognition—the neuronal spiking statistics, the effects of spiking synchronization by different brain waves, the roles played by synaptic efficacies and so forth. In particular, it is possible to demonstrate that networks with plastic and transient synaptic architectures can encode stable cognitive maps, revealing the characteristic timescales of memory processing.

Structuring Knowledge with Cognitive Maps and Cognitive Graphs

Humans and animals use mental representations of the spatial structure of the world to navigate. The classical view is that these representations take the form of Euclidean cognitive maps, but alternative theories suggest that they are cognitive graphs consisting of locations connected by paths. We review evidence suggesting that both map-like and graph-like representations exist in the mind/brain that rely on partially overlapping neural systems. Maps and graphs can operate simultaneously or separately, and they may be applied to both spatial and nonspatial knowledge. By providing structural frameworks for complex information, cognitive maps and cognitive graphs may provide fundamental organizing schemata that allow us to navigate in physical, social, and conceptual spaces.

Time as the fourth dimension in the hippocampus

Experiences of animal and human beings are structured by the continuity of space and time coupled with the uni-directionality of time. In addition to its pivotal position in spatial processing and navigation, the hippocampal system also plays a central, multiform role in several types of temporal processing. These include timing and sequence learning, at scales ranging from meso-scales of seconds to macro-scales of minutes, hours, days and beyond, encompassing the classical functions of short term memory, working memory, long term memory, and episodic memories (comprised of information about when, what, and where). This review article highlights the principal findings and behavioral contexts of experiments in rats showing: 1) timing: tracking time during delays by hippocampal 'time cells' and during free behavior by hippocampal-afferent lateral entorhinal cortex ramping cells; 2) 'online' sequence processing: activity coding sequences of events during active behavior; 3) 'off-line' sequence replay: during quiescence or sleep, orderly reactivation of neuronal assemblies coding awake sequences. Studies in humans show neurophysiological correlates of episodic memory comparable to awake replay. Neural mechanisms are discussed, including ion channel properties, plateau and ramping potentials, oscillations of excitation and inhibition of population activity, bursts of high amplitude discharges (sharp wave ripples), as well as short and long term synaptic modifications among and within cell assemblies. Specifically conceived neural network models will suggest processes supporting the emergence of scalar properties (Weber's law), and include different classes of feedforward and recurrent network models, with intrinsic hippocampal coding for 'transitions' (sequencing of events or places).

Three brain states in the hippocampus and cortex

Contemporary brain research seeks to understand how cognition is reducible to neural activity. Crucially, much of this effort is guided by a scientific paradigm that views neural activity as essentially driven by external stimuli. In contrast, recent perspectives argue that this paradigm is by itself inadequate and that understanding patterns of activity intrinsic to the brain is needed to explain cognition. Yet, despite this critique, the stimulus-driven paradigm still dominates-possibly because a convincing alternative has not been clear. Here, we review a series of findings suggesting such an alternative. These findings indicate that neural activity in the hippocampus occurs in one of three brain states that have radically different anatomical, physiological, representational, and behavioral correlates, together implying different functional roles in cognition. This three-state framework also indicates that neural representations in the hippocampus follow a surprising pattern of organization at the timescale of ∼1 s or longer. Lastly, beyond the hippocampus, recent breakthroughs indicate three parallel states in the cortex, suggesting shared principles and brain-wide organization of intrinsic neural activity.

Children five-to-nine years old can use path integration to build a cognitive map without vision

Although spatial navigation competence improves greatly from birth to adulthood, different spatial memory capacities emerge at different ages. Here, we characterized the capacity of 5-9-year-old children to use path integration to build egocentric and allocentric spatial representations to navigate in their environment, and compared their performance with that of young adults. First, blindfolded participants were tested on their ability to return to a starting point after being led on straight and two-legged paths. This egocentric homing task comprising angular and linear displacements allowed us to evaluate path integration capacities in absence of external landmarks. Second, we evaluated whether participants could use path integration, in absence of visual information, to create an allocentric spatial representation to navigate along novel paths between objects, and thus demonstrate the ability to build a cognitive map of their environment. Ninety percent of the 5-9-year-old children could use path integration to create an egocentric representation of their journey to return to a starting point, but they were overall less precise than adults. Sixty-four percent of 5-9-year-old children were capable of using path integration to build a cognitive map enabling them to take shortcuts, and task performance was not dependent on age. Imprecisions in novel paths made by the children who built a cognitive map could be explained by poorer integration of the experienced turns during the learning phase, as well as greater individual variability. In sum, these findings demonstrate that 5-9-year-old children can use path integration to build a cognitive map in absence of visual information.

Transition Scale-Spaces: A Computational Theory for the Discretized Entorhinal Cortex

Although hippocampal grid cells are thought to be crucial for spatial navigation, their computational purpose remains disputed. Recently, they were proposed to represent spatial transitions and convey this knowledge downstream to place cells. However, a single scale of transitions is insufficient to plan long goal-directed sequences in behaviorally acceptable time.

Here, a scale-space data structure is suggested to optimally accelerate retrievals from transition systems, called transition scale-space (TSS). Remaining exclusively on an algorithmic level, the scale increment is proved to be ideally [Formula: see text] for biologically plausible receptive fields. It is then argued that temporal buffering is necessary to learn the scale-space online. Next, two modes for retrieval of sequences from the TSS are presented: top down and bottom up. The two modes are evaluated in symbolic simulations (i.e., without biologically plausible spiking neurons). Additionally, a TSS is used for short-cut discovery in a simulated Morris water maze. Finally, the results are discussed in depth with respect to biological plausibility, and several testable predictions are derived. Moreover, relations to other grid cell models, multiresolution path planning, and scale-space theory are highlighted. Summarized, reward-free transition encoding is shown here, in a theoretical model, to be compatible with the observed discretization along the dorso-ventral axis of the medial entorhinal cortex. Because the theoretical model generalizes beyond navigation, the TSS is suggested to be a general-purpose cortical data structure for fast retrieval of sequences and relational knowledge.

Source code for all simulations presented in this paper can be found at https://github.com/rochus/transitionscalespace .

Replays of spatial memories suppress topological fluctuations in cognitive map

The spiking activity of the hippocampal place cells plays a key role in producing and sustaining an internalized representation of the ambient space-a cognitive map. These cells do not only exhibit location-specific spiking during navigation, but also may rapidly replay the navigated routs through endogenous dynamics of the hippocampal network. Physiologically, such reactivations are viewed as manifestations of "memory replays" that help to learn new information and to consolidate previously acquired memories by reinforcing synapses in the parahippocampal networks. Below we propose a computational model of these processes that allows assessing the effect of replays on acquiring a robust topological map of the environment and demonstrate that replays may play a key role in stabilizing the hippocampal representation of space.

Topological exploration of artificial neuronal network dynamics

One of the paramount challenges in neuroscience is to understand the dynamics of individual neurons and how they give rise to network dynamics when interconnected. Historically, researchers have resorted to graph theory, statistics, and statistical mechanics to describe the spatiotemporal structure of such network dynamics. Our novel approach employs tools from algebraic topology to characterize the global properties of network structure and dynamics. We propose a method based on persistent homology to automatically classify network dynamics using topological features of spaces built from various spike train distances. We investigate the efficacy of our method by simulating activity in three small artificial neural networks with different sets of parameters, giving rise to dynamics that can be classified into four regimes. We then compute three measures of spike train similarity and use persistent homology to extract topological features that are fundamentally different from those used in traditional methods. Our results show that a machine learning classifier trained on these features can accurately predict the regime of the network it was trained on and also generalize to other networks that were not presented during training. Moreover, we demonstrate that using features extracted from multiple spike train distances systematically improves the performance of our method.

A novel approach to model traffic on road segments of large-scale urban road networks

The study proposes a novel method for modeling traffic volumes at the road segment level of large-scale urban road networks. This study has been placed in a milieu where existing methods on modeling vehicular traffic volume are hampered by data and cost constraints, especially in developing countries. Emerging traffic modeling methods, based on centrality and space syntax provides a technically-efficient approach to overcome the above-mentioned constraints. Nevertheless, those methods are yet to be popular among practitioners due to limited accuracy and validity. This study modifies the existing methods and validates in five case cities to make them practice-ready. Findings of this study indicated that the proposed method is competent enough to estimate traffic volume of road segments on a par with the internationally accepted standards.

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The proposed method combines two network centrality measures abstracting the traffic volume on a road segment as the sum of origin-destination trips (i.e., Closeness-Centrality) and pass-by trips (i.e., Betweenness-Centrality).

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The study modifies the ‘distance’ variable in the existing formula as 'path-distance' which captures topological and mobility characteristics of roads.

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The method does not require extensive data and can be implemented by utilizing publicly available open-source network analysis software, hence, ideal for resource-scarce situations.

Navigating cognition: Spatial codes for human thinking

The hippocampal formation has long been suggested to underlie both memory formation and spatial navigation. We discuss how neural mechanisms identified in spatial navigation research operate across information domains to support a wide spectrum of cognitive functions. In our framework, place and grid cell population codes provide a representational format to map variable dimensions of cognitive spaces. This highly dynamic mapping system enables rapid reorganization of codes through remapping between orthogonal representations across behavioral contexts, yielding a multitude of stable cognitive spaces at different resolutions and hierarchical levels. Action sequences result in trajectories through cognitive space, which can be simulated via sequential coding in the hippocampus. In this way, the spatial representational format of the hippocampal formation has the capacity to support flexible cognition and behavior.

Non-Euclidean navigation

A basic set of navigation strategies supports navigational tasks ranging from homing to novel detours and shortcuts. To perform these last two tasks, it is generally thought that humans, mammals and perhaps some insects possess Euclidean cognitive maps, constructed on the basis of input from the path integration system. In this article, I review the rationale and behavioral evidence for this metric cognitive map hypothesis, and find it unpersuasive: in practice, there is little evidence for truly novel shortcuts in animals, and human performance is highly unreliable and biased by environmental features. I develop the alternative hypothesis that spatial knowledge is better characterized as a labeled graph: a network of paths between places augmented with local metric information. What distinguishes such a cognitive graph from a metric cognitive map is that this local information is not embedded in a global coordinate system, so spatial knowledge is often geometrically inconsistent. Human path integration appears to be better suited to piecewise measurements of path lengths and turn angles than to building a consistent map. In a series of experiments in immersive virtual reality, we tested human navigation in non-Euclidean environments and found that shortcuts manifest large violations of the metric postulates. The results are contrary to the Euclidean map hypothesis and support the cognitive graph hypothesis. Apparently Euclidean behavior, such as taking novel detours and approximate shortcuts, can be explained by the adaptive use of non-Euclidean strategies.

Through synapses to spatial memory maps via a topological model

Various neurophysiological and cognitive functions are based on transferring information between spiking neurons via a complex system of synaptic connections. In particular, the capacity of presynaptic inputs to influence the postsynaptic outputs–the efficacy of the synapses–plays a principal role in all aspects of hippocampal neurophysiology. However, a direct link between the information processed at the level of individual synapses and the animal’s ability to form memories at the organismal level has not yet been fully understood. Here, we investigate the effect of synaptic transmission probabilities on the ability of the hippocampal place cell ensembles to produce a cognitive map of the environment. Using methods from algebraic topology, we find that weakening synaptic connections increase spatial learning times, produce topological defects in the large-scale representation of the ambient space and restrict the range of parameters for which place cell ensembles are capable of producing a map with correct topological structure. On the other hand, the results indicate a possibility of compensatory phenomena, namely that spatial learning deficiencies may be mitigated through enhancement of neuronal activity.

Decoding of Neural Data Using Cohomological Feature Extraction

We introduce a novel data-driven approach to discover and decode features in the neural code coming from large population neural recordings with minimal assumptions, using cohomological feature extraction. We apply our approach to neural recordings of mice moving freely in a box, where we find a circular feature. We then observe that the decoded value corresponds well to the head direction of the mouse. Thus, we capture head direction cells and decode the head direction from the neural population activity without having to process the mouse's behavior. Interestingly, the decoded values convey more information about the neural activity than the tracked head direction does, with differences that have some spatial organization. Finally, we note that the residual population activity, after the head direction has been accounted for, retains some low-dimensional structure that is correlated with the speed of the mouse.

Robust spatial memory maps encoded by networks with transient connections

The spiking activity of principal cells in mammalian hippocampus encodes an internalized neuronal representation of the ambient space—a cognitive map. Once learned, such a map enables the animal to navigate a given environment for a long period. However, the neuronal substrate that produces this map is transient: the synaptic connections in the hippocampus and in the downstream neuronal networks never cease to form and to deteriorate at a rapid rate. How can the brain maintain a robust, reliable representation of space using a network that constantly changes its architecture? We address this question using a computational framework that allows evaluating the effect produced by the decaying connections between simulated hippocampal neurons on the properties of the cognitive map. Using novel Algebraic Topology techniques, we demonstrate that emergence of stable cognitive maps produced by networks with transient architectures is a generic phenomenon. The model also points out that deterioration of the cognitive map caused by weakening or lost connections between neurons may be compensated by simulating the neuronal activity. Lastly, the model explicates the importance of the complementary learning systems for processing spatial information at different levels of spatiotemporal granularity.

The reliability of our memories is nothing short of remarkable. Synaptic connections between neurons appear and disappear at a rapid rate, and the resulting networks constantly change their architecture due to various forms of neural plasticity. How can the brain develop a reliable representation of the world, learn and retain memories despite, or perhaps due to, such complex dynamics? Below we address these questions by modeling mechanisms of spatial learning in the hippocampal network, using novel algebraic topology methods. We demonstrate that although the functional units of the hippocampal network—the place cell assemblies—are unstable structures that may appear and disappear, the spatial memory map produced by a sufficiently large population of such assemblies robustly captures the topological structure of the environment.

Hexagonal Grid Fields Optimally Encode Transitions in Spatiotemporal Sequences

Grid cells of the rodent entorhinal cortex are essential for spatial navigation. Although their function is commonly believed to be either path integration or localization, the origin or purpose of their hexagonal firing fields remains disputed. Here they are proposed to arise as an optimal encoding of transitions in sequences. First, storage requirements for transitions in general episodic sequences are examined using propositional logic and graph theory. Subsequently, transitions in complete metric spaces are considered under the assumption of an ideal sampling of an input space. It is shown that memory capacity of neurons that have to encode multiple feasible spatial transitions is maximized by a hexagonal pattern. Grid cells are proposed to encode spatial transitions in spatiotemporal sequences, with the entorhinal-hippocampal loop forming a multitransition system.

Human spatial representation: what we cannot learn from the studies of rodent navigation

Studies of human and rodent navigation often reveal a remarkable cross-species similarity between the cognitive and neural mechanisms of navigation. Such cross-species resemblance often overshadows some critical differences between how humans and nonhuman animals navigate. In this review, I propose that a navigation system requires both a storage system (i.e., representing spatial information) and a positioning system (i.e., sensing spatial information) to operate. I then argue that the way humans represent spatial information is different from that inferred from the cellular activity observed during rodent navigation. Such difference spans the whole hierarchy of spatial representation, from representing the structure of an environment to the representation of subregions of an environment, routes and paths, and the distance and direction relative to a goal location. These cross-species inconsistencies suggest that what we learn from rodent navigation does not always transfer to human navigation. Finally, I argue for closing the loop for the dominant, unidirectional animal-to-human approach in navigation research so that insights from behavioral studies of human navigation may also flow back to shed light on the cellular mechanisms of navigation for both humans and other mammals (i.e., a human-to-animal approach).

Multidimensional brain activity dictated by winner-take-all mechanisms

A novel demon-based architecture is introduced to elucidate brain functions such as pattern recognition during human perception and mental interpretation of visual scenes. Starting from the topological concepts of invariance and persistence, we introduce a Selfridge pandemonium variant of brain activity that takes into account a novel feature, namely, demons that recognize short straight-line segments, curved lines and scene shapes, such as shape interior, density and texture. Low-level representations of objects can be mapped to higher-level views (our mental interpretations): a series of transformations can be gradually applied to a pattern in a visual scene, without affecting its invariant properties. This makes it possible to construct a symbolic multi-dimensional representation of the environment. These representations can be projected continuously to an object that we have seen and continue to see, thanks to the mapping from shapes in our memory to shapes in Euclidean space. Although perceived shapes are 3-dimensional (plus time), the evaluation of shape features (volume, color, contour, closeness, texture, and so on) leads to n-dimensional brain landscapes. Here we discuss the advantages of our parallel, hierarchical model in pattern recognition, computer vision and biological nervous system's evolution.

Uncovering temporal structure in hippocampal output patterns

Place cell activity of hippocampal pyramidal cells has been described as the cognitive substrate of spatial memory. Replay is observed during hippocampal sharp-wave-ripple-associated population burst events (PBEs) and is critical for consolidation and recall-guided behaviors. PBE activity has historically been analyzed as a phenomenon subordinate to the place code. Here, we use hidden Markov models to study PBEs observed in rats during exploration of both linear mazes and open fields. We demonstrate that estimated models are consistent with a spatial map of the environment, and can even decode animals’ positions during behavior. Moreover, we demonstrate the model can be used to identify hippocampal replay without recourse to the place code, using only PBE model congruence. These results suggest that downstream regions may rely on PBEs to provide a substrate for memory. Additionally, by forming models independent of animal behavior, we lay the groundwork for studies of non-spatial memory.

Still searching for the engram

For nearly a century, neurobiologists have searched for the engram-the neural representation of a memory. Early studies showed that the engram is widely distributed both within and across brain areas and is supported by interactions among large networks of neurons. Subsequent research has identified engrams that support memory within dedicated functional systems for habit learning and emotional memory, but the engram for declarative memories has been elusive. Nevertheless, recent years have brought progress from molecular biological approaches that identify neurons and networks that are necessary and sufficient to support memory, and from recording approaches and population analyses that characterize the information coded by large neural networks. These new directions offer the promise of revealing the engrams for episodic and semantic memories.

Hippocampus, Retrosplenial and Parahippocampal Cortices Encode Multicompartment 3D Space in a Hierarchical Manner

Humans commonly operate within 3D environments such as multifloor buildings and yet there is a surprising dearth of studies that have examined how these spaces are represented in the brain. Here, we had participants learn the locations of paintings within a virtual multilevel gallery building and then used behavioral tests and fMRI repetition suppression analyses to investigate how this 3D multicompartment space was represented, and whether there was a bias in encoding vertical and horizontal information. We found faster response times for within-room egocentric spatial judgments and behavioral priming effects of visiting the same room, providing evidence for a compartmentalized representation of space. At the neural level, we observed a hierarchical encoding of 3D spatial information, with left anterior hippocampus representing local information within a room, while retrosplenial cortex, parahippocampal cortex, and posterior hippocampus represented room information within the wider building. Of note, both our behavioral and neural findings showed that vertical and horizontal location information was similarly encoded, suggesting an isotropic representation of 3D space even in the context of a multicompartment environment. These findings provide much-needed information about how the human brain supports spatial memory and navigation in buildings with numerous levels and rooms.

Topological Schemas of Memory Spaces

Hippocampal cognitive map—a neuronal representation of the spatial environment—is widely discussed in the computational neuroscience literature for decades. However, more recent studies point out that hippocampus plays a major role in producing yet another cognitive framework—the memory space—that incorporates not only spatial, but also non-spatial memories. Unlike the cognitive maps, the memory spaces, broadly understood as “networks of interconnections among the representations of events,” have not yet been studied from a theoretical perspective. Here we propose a mathematical approach that allows modeling memory spaces constructively, as epiphenomena of neuronal spiking activity and thus to interlink several important notions of cognitive neurophysiology. First, we suggest that memory spaces have a topological nature—a hypothesis that allows treating both spatial and non-spatial aspects of hippocampal function on equal footing. We then model the hippocampal memory spaces in different environments and demonstrate that the resulting constructions naturally incorporate the corresponding cognitive maps and provide a wider context for interpreting spatial information. Lastly, we propose a formal description of the memory consolidation process that connects memory spaces to the Morris' cognitive schemas-heuristic representations of the acquired memories, used to explain the dynamics of learning and memory consolidation in a given environment. The proposed approach allows evaluating these constructs as the most compact representations of the memory space's structure.

3-D Maps and Compasses in the Brain

The world has a complex, three-dimensional (3-D) spatial structure, but until recently the neural representation of space was studied primarily in planar horizontal environments. Here we review the emerging literature on allocentric spatial representations in 3-D and discuss the relations between 3-D spatial perception and the underlying neural codes. We suggest that the statistics of movements through space determine the topology and the dimensionality of the neural representation, across species and different behavioral modes. We argue that hippocampal place-cell maps are metric in all three dimensions, and might be composed of 2-D and 3-D fragments that are stitched together into a global 3-D metric representation via the 3-D head-direction cells. Finally, we propose that the hippocampal formation might implement a neural analogue of a Kalman filter, a standard engineering algorithm used for 3-D navigation.

Interacting networks of brain regions underlie human spatial navigation: a review and novel synthesis of the literature

Navigation is an inherently dynamic and multimodal process, making isolation of the unique cognitive components underlying it challenging. The assumptions of much of the literature on human spatial navigation are that 1) spatial navigation involves modality independent, discrete metric representations (i.e., egocentric vs. allocentric), 2) such representations can be further distilled to elemental cognitive processes, and 3) these cognitive processes can be ascribed to unique brain regions. We argue that modality-independent spatial representations, instead of providing exact metrics about our surrounding environment, more often involve heuristics for estimating spatial topology useful to the current task at hand. We also argue that egocentric (body centered) and allocentric (world centered) representations are better conceptualized as involving a continuum rather than as discrete. We propose a neural model to accommodate these ideas, arguing that such representations also involve a continuum of network interactions centered on retrosplenial and posterior parietal cortex, respectively. Our model thus helps explain both behavioral and neural findings otherwise difficult to account for with classic models of spatial navigation and memory, providing a testable framework for novel experiments.

Exploring the alpha desynchronization hypothesis in resting state networks with intracranial electroencephalography and wiring cost estimates

This paper addresses a fundamental question, are eyes closed and eyes open resting states equivalent baseline conditions, or do they have consistently different electrophysiological signatures? We compare the functional connectivity patterns in an eyes closed resting state with an eyes open resting state to investigate the alpha desynchronization hypothesis. The change in functional connectivity from eyes closed to eyes open, is here, for the first time, studied with intracranial recordings. We perform network connectivity analysis in iEEG and we find that phase-based connectivity is sensitive to the transition from eyes closed to eyes open only in interhemispheral and frontal electrodes. Power based connectivity, on the other hand, consistently discriminates between the two conditions in temporal and interhemispheral electrodes. Additionally, we provide a calculation for the wiring cost, defined in terms of the connectivity between electrodes weighted by distance. We find that the wiring cost variation from eyes closed to eyes open is sensitive to the eyes closed and eyes open conditions. We extend the standard network-based approach using the filtration method from algebraic topology which does not rely on the threshold selection problem. Both the wiring cost measure defined here and this novel methodology provide a new avenue for understanding the electrophysiology of resting state.

Characteristics of CA1 place fields in a complex maze with multiple choice points

For the sake of rigorous control of task variables, hippocampal place cells have been usually studied in relatively simple environments. To approach the situation of real-life navigation in an urban-like environment, we recorded CA1 place cells while rats performance a memory task in a "Townmaze" with two start locations, three alternate paths in the maze midsection, followed by a two-way choice that determined the trial outcome, access to a goal compartment. Further, to test the ability of place cells to update their spatial representation upon local changes in the environment while maintaining the integrity of the overall spatial map to allow effective navigation, we occasionally introduced barriers in the maze mid-section to force the rat to select a nonpreferred route. The "Townmaze" revealed many new interesting features of CA1 neurons. First, we found neurons with 3-5 fields that appear to represent segments on a single common route through the maze. Second, we found neurons with 3-5 fields similarly aligned along the longitudinal or transverse maze axis. Responses to the barriers were assessed separately near and far from the barriers. Appearance of new fields in response to the barriers took place almost exclusively only locally near the barrier, whereas in-field firing rate changes occurred throughout the maze. Further, field location changes did not correlate with the task performance, whereas firing rate changes did. These findings suggest that in a complex environment with blocked distal views, CA1 neurons code for the environment as sequences of significant nodes but are also capable of extracting and associating common elements across these sequences.

The cognitive map in humans: spatial navigation and beyond

The 'cognitive map' hypothesis proposes that brain builds a unified representation of the spatial environment to support memory and guide future action. Forty years of electrophysiological research in rodents suggest that cognitive maps are neurally instantiated by place, grid, border and head direction cells in the hippocampal formation and related structures. Here we review recent work that suggests a similar functional organization in the human brain and yields insights into how cognitive maps are used during spatial navigation. Specifically, these studies indicate that (i) the human hippocampus and entorhinal cortex support map-like spatial codes, (ii) posterior brain regions such as parahippocampal and retrosplenial cortices provide critical inputs that allow cognitive maps to be anchored to fixed environmental landmarks, and (iii) hippocampal and entorhinal spatial codes are used in conjunction with frontal lobe mechanisms to plan routes during navigation. We also discuss how these three basic elements of cognitive map based navigation-spatial coding, landmark anchoring and route planning-might be applied to nonspatial domains to provide the building blocks for many core elements of human thought.

A roadmap for the computation of persistent homology

Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates, and provides a compact representation of the qualitative features of the input. The computation of PH is an open area with numerous important and fascinating challenges. The field of PH computation is evolving rapidly, and new algorithms and software implementations are being updated and released at a rapid pace. The purposes of our article are to (1) introduce theory and computational methods for PH to a broad range of computational scientists and (2) provide benchmarks of state-of-the-art implementations for the computation of PH. We give a friendly introduction to PH, navigate the pipeline for the computation of PH with an eye towards applications, and use a range of synthetic and real-world data sets to evaluate currently available open-source implementations for the computation of PH. Based on our benchmarking, we indicate which algorithms and implementations are best suited to different types of data sets. In an accompanying tutorial, we provide guidelines for the computation of PH. We make publicly available all scripts that we wrote for the tutorial, and we make available the processed version of the data sets used in the benchmarking.

ELECTRONIC SUPPLEMENTARY MATERIAL

The online version of this article (doi:10.1140/epjds/s13688-017-0109-5) contains supplementary material.