Clique topology reveals intrinsic geometric structure in neural correlations

Detecting meaningful structure in neural activity and connectivity data is challenging in the presence of hidden nonlinearities, where traditional eigenvalue-based methods may be misleading. We introduce a novel approach to matrix analysis, called clique topology, that extracts features of the data invariant under nonlinear monotone transformations. These features can be used to detect both random and geometric structure, and depend only on the relative ordering of matrix entries. We then analyzed the activity of pyramidal neurons in rat hippocampus, recorded while the animal was exploring a two-dimensional environment, and confirmed that our method is able to detect geometric organization using only the intrinsic pattern of neural correlations. Remarkably, we found similar results during non-spatial behaviors such as wheel running and REM sleep. This suggests that the geometric structure of correlations is shaped by the underlying hippocampal circuits, and is not merely a consequence of position coding. We propose that clique topology is a powerful new tool for matrix analysis in biological settings, where the relationship of observed quantities to more meaningful variables is often nonlinear and unknown.Comment: 29 pages, 4 figures, 13 supplementary figures (last two authors contributed equally

Understanding short-timescale neuronal firing sequences via bias matrices

The brain generates persistent neuronal firing sequences across varying timescales. The short-timescale (~100ms) sequences are believed to be crucial in the formation and transfer of memories. Large-amplitude local field potentials known as sharp-wave ripples (SWRs) occur irregularly in hippocampus when an animal has minimal interaction with its environment, such as during resting, immobility, or slow-wave sleep. SWRs have been long hypothesized to play a critical role in transferring memories from the hippocampus to the neocortex [1]. While sequential firing during SWRs is known to be biased by the previous experiences of the animal, the exact relationship of the short-timescale sequences during SWRs and longer-timescale sequences during spatial and nonspatial behaviors is still poorly understood. One hypothesis is that the sequences during SWRs are “replays” or “preplays” of “master sequences”, which are sequences that closely mimic the order of place fields on a linear track [2,3]. Rather than particular hard-coded “master” sequences, an alternative explanation of the observed correlations is that similar sequences arise naturally from the intrinsic biases of firing between pairs of cells. To distinguish these and other possibilities, one needs mathematical tools beyond the center-of-mass sequences and Spearman’s rank-correlation coefficient that are currently used

Allothetic and idiothetic spatial tasks in rats and human.

Available from STL Prague, CZ / NTK - National Technical LibrarySIGLECZCzech Republi

Cell Assembly Sequences Arising from Spike Threshold Adaptation Keep Track of Time in the Hippocampus

Hippocampal neurons can display reliable and long-lasting sequences of transient firing patterns, even in the absence of changing external stimuli. We suggest that time-keeping is an important function of these sequences, and propose a network mechanism for their generation. We show that sequences of neuronal assemblies recorded from rat hippocampal CA1 pyramidal cells can reliably predict elapsed time (15–20 s) during wheel running with a precision of 0.5 s. In addition, we demonstrate the generation of multiple reliable, long-lasting sequences in a recurrent network model. These sequences are generated in the presence of noisy, unstructured inputs to the network, mimicking stationary sensory input. Identical initial conditions generate similar sequences, whereas different initial conditions give rise to distinct sequences. The key ingredients responsible for sequence generation in the model are threshold-adaptation and a Mexican-hat-like pattern of connectivity among pyramidal cells. This pattern may arise from recurrent systems such as the hippocampal CA3 region or the entorhinal cortex.Wehypothesize that mechanisms that evolved for spatial navigation also support tracking of elapsed time in behaviorally relevant contexts

Understanding short-timescale neuronal firing sequences via bias matrices

The brain generates persistent neuronal firing sequences across varying timescales. The short-timescale (~100ms) sequences are believed to be crucial in the formation and transfer of memories. Large-amplitude local field potentials known as sharp-wave ripples (SWRs) occur irregularly in hippocampus when an animal has minimal interaction with its environment, such as during resting, immobility, or slow-wave sleep. SWRs have been long hypothesized to play a critical role in transferring memories from the hippocampus to the neocortex [1]. While sequential firing during SWRs is known to be biased by the previous experiences of the animal, the exact relationship of the short-timescale sequences during SWRs and longer-timescale sequences during spatial and nonspatial behaviors is still poorly understood. One hypothesis is that the sequences during SWRs are “replays” or “preplays” of “master sequences”, which are sequences that closely mimic the order of place fields on a linear track [2,3]. Rather than particular hard-coded “master” sequences, an alternative explanation of the observed correlations is that similar sequences arise naturally from the intrinsic biases of firing between pairs of cells. To distinguish these and other possibilities, one needs mathematical tools beyond the center-of-mass sequences and Spearman’s rank-correlation coefficient that are currently used

Pairwise correlation graphs from hippocampal population activity have highly non-random, low-dimensional clique topology

We analyzed the structure of cliques in pairwise correlation graphs obtained from population activity in hippocampus under a variety of behavioral conditions: open field exploration, wheel running, and sleep. Using topological data analysis, we found these graphs to be highly non-random by comparing their clique structure to that of Erdös-Rényi random graphs with matching edge probabilities. The clique topology in all three conditions was low-dimensional and consistent with what is theoretically expected from place cells during spatial navigation. Remarkably, the same pattern was also observed in the sleep data. To better understand how this correlation structure might arise, we considered a recurrent network with a simple Hebbian learning rule. We found the same clique structure emerged in the network when recurrent connections were formed in the presence of random sparse patterns of activity. This may provide a generic mechanism explaining how low-dimensional clique topology arises in hippocampal data. Pairwise correlations are an important tool for understanding neuronal population activity [1]. Pairwise correlation graphs, where the edges reflect high levels of correlation between neurons, are often used as a proxy for underlying network connectivity. In this work, we are motivated by the question: What is the structure of pairwise correlations in hippocampal population activity? To address this question, we analyzed graded families of pairwise correlation graphs, parametrized by the threshold on correlation strength used to define the edges. In brain areas with receptive fields or place fields, the structure of the neural code has strong implications for structure of cliques in these graphs. This can be detected by examining the clique topology of the graph (specifically, topological invariants called ‘Betti numbers’ of the clique complex), and is closely tied to the dimension and topology of the underlying space [2]. For example, for hippocampal place cell activity during spatial exploration, pairwise correlation graphs are expected to have highly non-random and lowdimensional clique topology, due to the arrangement of place fields in a low-dimensional environment

Wang_et_al_eLife2016_data_part4

Summary The DRYAD data sets Wang_et_al_eLife2016_data contain neuronal data recorded by two 64-channel silicon probes from CA1 hippocampal regions of rats that were performing a delayed left/right arm alternation task. Some recordings were performed after medial septum inactivation (see Wang et al., Nature Neuroscience 2015 for details). The data structures include: • Wave-forms of putative spikes extracted from original raw broadband signals • LFPs (local field potentials) • Results of spike sorting • Information about the animal behavior during the experiment Data files Wang_et_al_eLife2016_data_part4.tar: A543-20120412-03_BehavElectrDataLFP.mat A543-20120422-03_BehavElectrDataLFP.mat A543-20120425-02_BehavElectrDataLFP.mat A943-20120515-03_BehavElectrDataLFP.mat A943-20120521-03_BehavElectrDataLFP.mat A943-20120523-03_BehavElectrDataLFP.mat A943-20120526-03_BehavElectrDataLFP.ma

Wang_et_al_eLife2016_data_part2.tar

Summary The DRYAD data sets Wang_et_al_eLife2016_data contain neuronal data recorded by two 64-channel silicon probes from CA1 hippocampal regions of rats that were performing a delayed left/right arm alternation task. Some recordings were performed after medial septum inactivation (see Wang et al., Nature Neuroscience 2015 for details). The data structures include: • Wave-forms of putative spikes extracted from original raw broadband signals • LFPs (local field potentials) • Results of spike sorting • Information about the animal behavior during the experiment Data files Control conditions (before medial septum inactivation) Wang_et_al_eLife2016_data_part2.tar: A543-20120412-01_BehavElectrDataLFP.mat A543-20120422-01_BehavElectrDataLFP.mat A543-20120425-01_BehavElectrDataLFP.mat A943-20120515-01_BehavElectrDataLFP.mat A943-20120521-01_BehavElectrDataLFP.mat A943-20120523-01_BehavElectrDataLFP.mat A943-20120526-01_BehavElectrDataLFP.ma

Wang_et_al_eLife2016_data_part3

Summary The DRYAD data sets Wang_et_al_eLife2016_data contain neuronal data recorded by two 64-channel silicon probes from CA1 hippocampal regions of rats that were performing a delayed left/right arm alternation task. Some recordings were performed after medial septum inactivation (see Wang et al., Nature Neuroscience 2015 for details). The data structures include: • Wave-forms of putative spikes extracted from original raw broadband signals • LFPs (local field potentials) • Results of spike sorting • Information about the animal behavior during the experiment Data files Experimental conditions (after medial septum inactivation) Wang_et_al_eLife2016_data_part3.tar: A498-20120807-04_BehavElectrDataLFP.mat A498-20120809-05_BehavElectrDataLFP.mat A498-20120813-02_BehavElectrDataLFP.mat A498-20120815-03_BehavElectrDataLFP.mat A498-20120827-03_BehavElectrDataLFP.ma