Inverse Network Dynamics - Network structure and function from nonlinear dynamics and time series data

Reik V. Donner (1,2), Yong Zou (3), Frederik Wolf (2), Adrian Odenweller (2) (1) Magdeburg-Stendal University of Applied Sciences, Magdeburg, Germany (2) Potsdam Institute for Climate Impact Research (PIK) – Member of the Leibniz Association, Potsdam, Germany (3) East China Normal University, Shanghai, China During the last two decades, functional network representations generated from multivariate time series have become an important tool for studying dynamical processes in complex systems like the human brain or the Earth’s climate system. By exploiting statistical associations among the time series, the presence of strong statistical similarity between pairs of series is taken as a proxy of a functional relationship between the subsystems represented by the two series, which may eventually be attributed to specific physical processes. In most classical applications, the data used for functional network inference are continuous time series with regular spacing of individual observations. In such situations, statistical similarity is typically quantified by corresponding association measures like Pearson correlation or mutual information. However, there also exist relevant applications of functional networks where the underlying data solely contain the timing of discrete events like neuronal spikes in the epileptic brain, heavy rainfall events or earthquakes, to mention only a few examples. In such cases, techniques quantifying the synchrony of the event sequences serve as proper similarity measures upon which the network construction can be based. Since the introduction of the associated concept of event synchronization concept 20 years ago, a multiplicity of different statistics has been developed and applied in different application contexts. Recent results for functional climate networks of heavy precipitation events as well as networks constructed from numerical spreading models have demonstrated that two of the most widely applied association measures for event sequences, event synchronization strength and event coincidence rate, tend to differ in the network characteristics derived therefrom, despite following a largely identical conceptual approach. One main reason for the corresponding discrepancies in climate applications originates when events occur clustered in time, calling for a careful definition of events when employing the event synchronization method to such types of data. One potential benefit of event based similarity measures is that they offer a natural way for identifying statistical directionality in the interrelationship between two series, since they are commonly not exactly symmetric under exchange of the series as it would be the case for lag-zero correlations and other measures based on continuous variability. However, was has pertained so far is the problem of distinguishing direct from indirect linkages when resorting only to bivariate association measures. As a first step forward from mere event coincidence to actual causality that allows distinguishing direct from indirect relationships in functional network construction, the established “partialization trick” previously used in phase synchronization contexts, which is based on the inverse matrix of pairwise association measures, can be used to define partial event coincidence rates. Results for networks of coupled chaotic oscillators as well as resting state EEG data demonstrate the great potential of this approach for generating functional network representations that contain solely those statistical associations that cannot be explained by the action of any third variable.

In our study, computer simulations of a simple dynamical model of the brain are presented. The model uses an empirical network structure obtained from Diffusion Tensor Imaging (DTI) data and generic dynamics on each node of the network. The model is employed to simulate brain flexibility patterns in order to have a way to assess the mechanism behind the flexibility measure [chinichian et al 2022]. We show that the brain connection structure and the collection of nodes that are being stimulated play crucial roles in the brain-like pattern observed.

In the analysis of oscillatory processes, one needs a good estimation of the phase. Here I discuss a recently suggested approach to phase demodulation based on iterated Hilbert transforms. I demonstrate that for purely phase modulated signals this method provides a perfect demodulation, while its performance is much more pure if also an amplitude modulation is presented. I also discuss, how a transformation from a generic protophase of an oscillatory signal to a ``true phase'' can be accomplished.

We study the dynamics of networks of phase oscillators with varying coupling strengths and connectivity between oscillators. Such adaptive systems have been used to model certain networks of oscillatory neurons. In neuronal networks, the spiking dynamics of the neurons, synaptic weights, and network structure evolve together, guiding one another. Certain neuronal networks exhibit coherent, incoherent, and clustered states, and sometimes multi-stability. We model weight adaptation using spike-timing-dependent plasticity and structural plasticity as stochastic addition and elimination of connections. We show the coexistence of synchronized and desynchronized states with identical oscillators and that of multiple states with non-identical oscillators. We show that structural plasticity optimizes the synchronized state of a network by significantly reorganizing the network structure and allowing for synchronization with lesser connectivity compared to a network with weight plasticity alone. In case of non-identical oscillators, the structural plasticity results in strong dependence of the connectivity of a given oscillator on its characteristic frequency. Furthermore, the structural plasticity strengthens both synchronized and desynchronized states of a network. We demonstrate that by using a desynchronizing Coordinated Reset stimulation and a synchronizing periodic stimulation. Our results indicate that a synchronized (desynchronized) network is easier to desynchronize (synchronize) in the absence of structural plasticity, confirming that the structural plasticity makes the dynamical states more stable.

We discuss several data analysis tasks appearing in the context of deep brain stimulation. The first task is a real-time estimation of the signal's phase and amplitude. The next problem is inferring the response characteristics of an oscillator exploiting test stimulation with pulses of realistic shape, particularly charge-balanced pulses used in neuroscience applications. We discuss the determination of the phase and amplitude response separately.

Brain dynamics naturally constitute one of the archetypal complex systems showing a plethora of rich emergent phenomena. To understand the mechanisms behind the observed properties, computational modeling and advanced data analysis approaches are increasingly adopted. However, the complexity of the applied approaches may lead into a new set of problems, related to the ambiguity of the appropriate interpretation of the analysis or modeling results. In such cases, it may be suitable to apply the general heuristic principle of parsimony, sometimes called Occam's razor. In this contribution we shall demonstrate, how some rich and complex properties of brain dynamics and connectivity structure can be explained from relatively simple principles and models (including linear stochastic dynamics), and which aspects remain to be explained by some richer model. The specific examples include the small-world property of brain connectivity networks [1], the temporal properties of of brain states [2] and assessing the utility of nonlinear network inference from particular datasets [3]. Relevance of such questions to analysis of other complex systems (climate dynamics, stock networks) will be also highlighted.

If vertices or edges of a networked dynamical system are not accessible, a widely-used approach is to resort to the functional network ansatz. Such a network is derived from analyzing properties of interactions between system units (vertices) to draw conclusions about the organization of the structural network. Despite some progress in this active field of research, there are still problems that evade from a satisfactory solution. We investigate if and to what extent important properties of a functional network on the global and the local scale match those of the corresponding structural network. We derive functional networks from characterizing interactions in paradigmatic oscillator networks using widely-used time-series analysis techniques. Surprisingly, we observe local characteristics of key constituents of functional networks, identified with different centrality concepts, to largely coincide with the structural networks' analogues, while global topological and spectral aspects deviate. Authors: Timo Bröhl, Klaus Lehnertz, Thorsten Rings

Currently, network becomes essential ingredient to study the underlying universal characteristics in several real-world situations and corresponding physical systems. For instance, connectivity pattern in cell cellular signal networks, people interaction in social media, essential elements of power girds: all can be mapped with several network properties. However, the physical phenomena are not fully inferable without using underlying dynamics working over the of the network and it is possible that the dynamical observables can be infer from the network topology. Here we propose a general scheme to identify the inner network symmetry with the help of eigen-vector centrality. We unveil that there is a direct connection between the elements of eigen-vector centrality and the clusters of a network. P. Khanra, S. Ghosh, K. Alfaro-Bittner, P. Kundu, S. Boccaletti, C. Hens, and P. Pal, identifying symmetries and predicting cluster synchronization in complex networks, Chaos, Solitons Fractals 155, 111703 (2022).

In the fields of complex dynamics and complex networks, the inverse problem is generally regarded as hard and extremely challenging mathematically as complex dynamical systems and networks consists of a large number of interacting units. However, our ideas based on compressive sensing, in combination with innovative approaches, generates a new paradigm that offers the possibility to address the fundamental inverse problem in complex dynamics and networks. In particular, in this talk, I will argue that evolutionary games model, a common type of interactions in a variety of complex, networked, natural systems and social systems, allows the uncovering of the interacting structure of the underlying network and the understanding of its collective dynamics from small amounts of data. The method is validated by conducting an actual experiment to reconstruct a social network. Based on ecological real-world networks, I will discuss the interplay between transients and stochasticity in empirical mutualistic networks. Focusing on the tipping-point dynamics, I will discuss the phenomena of noise-induced collapse and noise-induced recovery. Two types of noise are considered, environmental (Gaussian white) noise and state-dependent demographic noise. I will also discuss control strategies that delay the extinction and advances the recovery by controlling the decay rate of pollinators in the stochastic mutualistic complex network. The phenomena of noise-induced collapse and recovery and the associated scaling laws, and the control of tipping point strategies have implications to managing high-dimensional ecological systems. ----------- Data based identification and prediction of nonlinear and complex dynamical systems, W.-X. Wang, Y.-C. Lai, and C. Grebogi, Phys. Reports 644, 1-76 (2016) Predicting catastrophe in nonlinear dynamical systems by compressive sensing, W.-X. Wang, R. Yang, Y.-C. Lai, V. Kovanis, and C. Grebogi, Phys. Rev. Lett. 106, 154101 (2011) Network reconstruction based on evolutionary-game data via compressive sensing, W.-X. Wang, Y.-C. Lai, C. Grebogi, and J. Ye, Phys. Rev. X 1, 021021 (2011) Predicting tipping points in mutualistic networks through dimension reduction, J. Jiang, Z.-G. Huang, T.P. Seager, W. Lin, C. Grebogi, A. Hastings, and Y.-C. Lai, PNAS (Proc. Nat. Acad. Sci.) 115, E639-E647 (2018) Noise-enabled species recovery in the aftermath of a tipping point, Y. Meng, J. Jiang, C. Grebogi, and Y.-C. Lai, Phys. Rev. E 101, 012206 (2020) Tipping point and noise-induced transients in ecological networks, Y. Meng, Y.-C. Lai, and C. Grebogi, J. Royal Soc. Interface 17, 20200645 (2020) Control of tipping points in stochastic mutualistic complex networks, Y. Meng and C. Grebogi, Chaos 31, 023118 (2021) Sudden regime shifts after apparent stasis: Comment on long transients in Ecology, C. Grebogi, Physics of Life Reviews 32, 41 (2020)

Analysis of networks with multiple variables using causal inference tools has become very popular recently because of the development of mathematical tools and algorithms for causal discovery. For systems where temporal measurements are available, the pioneering concept of Granger Causality as well as its many extensions are being employed. These methods help to capture the interactions between network variables where exact model discovery is difficult. Contemporaneously, the field of compressed sensing (CS) has seen an increasing interest in the number and scope of its applications for the acquisition, storage and transmission of signals. While conventional signal acquisition protocols require that the signals are sampled at atleast the Nyquist frequency to keep the signal information intact, CS allows for the possibility of using far fewer number of measurements to acquire and represent the signal. The only requirement of CS is that the signal being sensed be `sparse’ in some domain, which is satisfied by many naturally occurring signals. Hence, CS has been adopted rapidly in the design of signal and image acquisition strategies, e.g., magnetic resonance imaging, scanning tunneling microscopy, quantum state tomography, analog to digital conversion technologies. Currently, the compressed signals are required to be reconstructed in order to make causal discovery analysis from compressively sensed signals possible. In this work, we provide a mathematical proof that structured compressed sensing matrices, specifically Circulant and Toeplitz, preserve Granger causality relationships in the compressed signal domain. This theorem is then verified on bivariate and multivariate sparse neuronal spike train simulations compressed using the recommended structured matrices. The approach is also applied to real neuronal spike trains from a rat prefrontal cortex to infer network causal connectivity. Demonstration of the possibility to make causal inferences in the compressed domain is useful to reduce computational-cost, to correctly apply the well-known Granger-Causality and to design appropriate sensing matrices. Authors: Aditi Kathpalia, Nithin Nagaraj Acknowledgements: This study is supported by the Czech Science Foundation, Project No. GA19-16066S and by the Czech Academy of Sciences, Praemium Academiae awarded to M. Paluš. Nithin Nagaraj gratefully acknowledges the financial support of Cognitive Science Research Initiative, Dept. of Science & Tech., Govt. of India (CSRI-DST) Grant No. DST/CSRI/2017/54(G).

Can a collection of disparate and severely-underdetermined systems of equations be solvable only when considered as a whole? We propose a novel mathematical approach to answer this question and show direct applications to challenging cornerstone problems within the network reconstruction field, such as (i) reconstruction under variability of network parameters across experiments, (ii) tracking of structural changes in networks, and (iii) purely data-driven reconstruction.

The reliable inference of networks from data remains a challenge, in particular when it comes to distinguishing direct and indirect interactions in large networks. While progress has been made, challenges still remain. This presentation aims to give an overview of techniques that promise to address the challenge of inferring network structures reliably in larger networks. Issues such as the presence of highly correlated components present in the networks will be discussed. Simulation studies as well as real world data examples will demonstrate abilities but also limitations of the suggested approaches.

One of the most challenging aspects in the study of the complex dynamical systems is the identification of their underlying interdependence structure from time series data. Being in the era of Big Data, this problem gets even more complicated since the number of available variables may be very large. Under this setting, the estimation of direct causality with a Granger causality-type measure inevitably has to involve some kind of dimension reduction. The measure should, ideally, also be capable to detect nonlinear effects, persistently present in real world complex systems. The model-free information-based measure of partial mutual information from mixed embedding (PMIME) [1] has been developed to address these issues and it was found to perform well on multivariate time series of moderately high dimension. Specifically, it was found that the estimated causality network by PMIME matches well the complex network of the interdependence structure of the original complex system [2]. In this presentation, I will review the PMIME measure and present some modifications on PMIME: 1) the inclusion of pairs of lag terms to address the presence of only synergetic effects, 2) the correction of the significance test for conditional mutual information used as termination criterion that renders the bias towards including less significant terms. Both modifications increase the computation time and the trade-off of accuracy and computation time will be demonstrated with synthetic examples. Further, the effect of the dimension of the system on the estimation accuracy of the original complex network will be investigated in relation to the sparsity of the original network. Examples from multichannel EEG analysis and finance will be given. References [1] D. Kugiumtzis, Physical Review E, 87 (2013) 6. [2] C. Koutlis, D. Kugiumtzis, Chaos, 26 (2016) 093120.

We consider approaches which deal with the effects of conductivity and reference electrode on measures of connectivity or synchronization used to extract brain functional networks from scalp EEG recordings. Then we focus on changes of network structure in time and ask whether there are some regular patterns or random fluctuations and discuss how to characterize changes in brain states due to mental disorders and/or their treatment. The support by the Czech Science Foundation (Project No. GF21-14727K) is gratefully acknowledged.

To what extent can activity at one network node be inferred by observing another? In real systems, observations are noisy, and this question will be decided by network geometry, and its effect on amplification of noise throughout the network. We propose a single quantity, depending on the desired node and the observing subset, that is able to characterize the relevant error magnification. This error magnification quantity could be helpful for deciding among alternatives for observing the network. We begin by surveying results from linear dynamics, and discuss similarities and differences with nonlinear network dynamics.

I will present a recent development in the probabilistic approach to Takens time-delay embeddings of dynamical systems. In this setting, one assumes that initial states of the system are generated randomly with a given distribution and one is interested in a unique reconstruction of a typical random initial point. It was conjectured by Shroer, Sauer, Ott and Yorke (Predicting chaos most of the time from embeddings with self-intersections. Phys. Rev. Lett. 80, 1410–1413 (1998)) that in the probabilistic setting, one can reduce the number of required measurements twice (with respect to the classical deterministic results). I will present probabilistic versions of the Takens theorem, which are theoretical confirmation of this conjecture. I will also explain what is known about regularity properties of predictions made in the probabilistic setting. The talk will be based on joint works with Krzysztof Barański and Yonatan Gutman.

Purely predictive methods do not perform well when the test distribution changes too much from the training distribution. Causal models are known to be stable with respect to several perturbations, but may not perform well when the test distribution differs only mildly from the training distribution. As a result, methods have been proposed that provide a trade-off between causal and predictive models. In this talk, we introduce CausalKinetiX, a framework that aims at learning stable structures in dynamical systems by trading off invariance and predictability. The talk contains joint work with Niklas Pfister and Stefan Bauer. No prior knowledge about causality is required.

The first part of this talk will report a machine learning (ML) method for inferring short term causal interactions to deduce network links [1], and will test this technique, combined with surrogate time series data methods, on C. elegans neuronal network data (for which the ground truth network is known). In the second part, we use link inference to enable ML prediction of the evolution of nodal network states [2] and discuss how the link inference and the nodal dynamics prediction tasks interact and compliment each other. [1] A. Banerjee et al., Chaos, 12104 (2019); Phys. Rev. X, 031014 (2021). [2] K. Srinivasan et al., Phys. Rev. Lett.(2022). Co-authors: A. Banerjee, K. Sreenivasan, S. Chandra, M. Girvan, R. Roy, J. Hart, T. M. Antonsen, N. Coble, J. Hamlin.

The experimental analysis of multiple biological systems relies on tracking reporters of their state over time. Here, I will discuss the survival statistics methods we have used to infer the properties of networks of molecular interactions in the disaggregation of pathological amyloid structures [1] and cellular processes in gene regulation through the cell cycle [2]. References: 1. A. Franco, P. Gracia, A. Colom, J. D. Camino, J. A. Fernandez-Higuero, N. Orozco, A. Dulebo, L. Saiz, N. Cremades, J. M. G. Vilar, A. Prado, A. Muga, All-or-none amyloid disassembly via chaperone-triggered fibril unzipping favors clearance of α-synuclein toxic species, Proc. Natl. Acad. Sci. USA 118, e2105548118 (2021). 2. C. Chang, M. Garcia-Alcala, L. Saiz, J.M.G. Vilar, and P. Cluzel, Robustness of DNA Looping Across Multiple Cell Divisions in Individual Bacteria, Proc. Natl. Acad. Sci. USA 119, e2200061119 (2022).

The sphera-corpus is a set of more than 350 editions of astronomic text books for students used for university teaching in the early modern times all across Europe. Considering each edition as a node, a multi-layer network is formed by the definition of links between books. Such links characterize different types of semantic and socio-economic relationships. Time resolved network analysis provides insights into the process of knowledge accumulation, diversification, and consolidation over a time span of more than 180 years. Particularly important editions in terms of innovation and influence on later editions have been identified. Specific information is obtained by using conditional sub-networks, i.e., removing links from single layer-subnetworks depending on the existence or lack of existence of the corresponding links in other layers. From such type of analysis, mechanisms for the propagation of knowledge through the corpus can be identified.

We demonstrate a network recovery approach to learning the connectivity structure of a complex network from time-series observations using dynamical systems and interpretable machine learning tools. In particular, we focus on neuron networks which are described by chaotic isolated dynamics [1], weakly interacting nodes [2] and interaction through scale-free type networks [3]. Network community structure is recovered accurately in this setting using a reduction theorem [4]. Our approach uses the same theory, determines all the unknowns one by one, and finally learns the exact connectivity structure with the $\ell_1$-norm. We apply our technique to weakly coupled Rulkov maps on a real mouse neocortex network as an implementation. [1] H. Korn and P. Faure, Is there chaos in the brain? ii. experimental evidence and related models, Comptes Rendus - Biologies 326, 787 (2003). [2] A. J. Preyer and R. J. Butera, Neuronal oscillators in aplysia californica that demonstrate weak coupling in vitro, Physical Review Letters 95, 10.1103/PhysRevLett.95.138103 (2005). [3] G. Werner, Fractals in the nervous system: Conceptual implications for theoretical neuroscience, Frontiers in Physiology 1 JUL, 10.3389/fphys.2010.00015 (2010). [4] D. Eroglu, M. Tanzi, S. van Strien, and T. Pereira, Revealing Dynamics, Communities, and Criticality from Data, Physical Review X 10, 1 (2020), ISSN 2160-3308.

What do generic networks that have certain properties look like? We use Relative Canonical Network Ensembles as the ensembles that realize a property R while being as indistinguishable as possible from a background network ensemble. This allows us to study the most generic features of the networks giving rise to the property under investigation.

In the past decades machine learning has had a rapidly growing impact on many fields of natural-, life- and social sciences as well as engineering. Machine learning excels at classification and regression tasks from complex heterogeneous datasets and can answer questions like "What statistical associations or correlations can we see in the data?", "What objects are in this picture?", or "What is the most likely next data point?". But many questions in science, engineering, and politics are about "What are the causal relations underlying the data?" or "What if a certain variable changes or is changed?" or "What would have happened if some variable had another value?". Data-driven machine learning alone fails to answer such questions. Causal inference provides the theory and methods to learn and utilize qualitative knowledge about causal relations. Together with machine learning it enables causal reasoning given complex data. Furthermore, causal methods can be used to intercompare and validate physical simulation models. In this talk I will present an overview of this exciting and widely applicable framework and illustrate it with some examples from Earth sciences and beyond.

Inferring the timing and amplitude of perturbations in epidemiological systems from their stochastically spread low-resolution outcomes is as relevant as challenging. It is a requirement for current approaches to overcome the need to know the details of the perturbations to proceed with the analyses. However, the general problem of connecting epidemiological curves with the underlying incidence lacks the highly effective methodology present in other inverse problems, such as super-resolution and dehazing from machine vision. I will present an unsupervised physics-informed convolutional neural network approach in reverse to connect death records with incidence that allows the identification of regime changes at a single-day resolution. Applied to COVID-19 data with proper regularization and model-selection criteria, the approach can identify the implementation and removal of lockdowns and other nonpharmaceutical interventions with ± 0.8-day accuracy over the time span of a year.

The ability to infer causal relations between observable phenomena is important in many scientific disciplines, however the means for such enterprise without experimental manipulation are limited. A commonly applied principle is that of the cause preceding and predicting the effect, taking into account other circumstances. While this 'taking into account' may pose a computational challenge calling for the construction of sophisticated inference algorithms [1], a much deeper problem arises when unobserved hidden causal variables are present. An interesting heuristic suggests that such latent variable effects should also present in time-reversed series, and may be corrected for by subtracting the causality index obtained in the reverse direction from time reversed data [2]. Indeed, intuitively, when the temporal order of events is reverted, one expects the cause and effect to switch roles. This has been demonstrated in bivariate linear systems and utilized in design of improved causal inference scores, and such behavior in linear systems has been put in contrast with nonlinear chaotic systems where the inferred causal direction appears unchanged under time reversal. We follow the observations in linear processes in more depth [3], extending the analysis to networks, deriving the necessary and sufficient condition of causal structure reversal under time reversal for vector autoregressive processes of order 1; showing that the key property is the normality of the causal interaction matrix. We provide an analytic proof that causal structure reversal happens perfectly in the bivariate case only in two particular trivial symmetric cases, and the minimal unidirectionally coupled network showing causal structure reversal has 3 subsystems. We finally turn to real-world data, studying the extent to which the causal structure reversal appears in brain and climate network under time reversal, showing that already the linear approximations of both brain and climate systems show imperfect causal structure reversal under time reversal, while the extent to which this property is broken is closely predicted by deviation of the coupling matrix from normality. Finally, we discuss the relevance of these findings in a wider context, including how it problematizes the use of comparison with time-reversed version of a given process for inference of causal structures. [1] J. Korenek and J. Hlinka. Causal network discovery by iterative conditioning: Comparison of algorithms. Chaos: An Interdisciplinary Journal of Nonlinear Science, 30(1):013117, 2020. [2] Winkler, I.; Panknin, D.; Bartz, D.; Muller, K.R.; Haufe, S. Validity of Time Reversal for Testing Granger Causality. IEEE Transactions on Signal Processing, 64, 2746–2760, 2016. [3] J. Korenek and J. Hlinka. Causality in reversed time series: Reversed or conserved? Entropy, 23(8):1067, 2021.

Ideas from Deep Learning have begun to impact different sciences, ranging from engineering to biology and medicine. Breakthroughs continue to make headlines, yielding data-driven models that can accurately translate speech to text, play games with superhuman performance, or generate photo-realistic images from text descriptions only. In this talk, I will overview recent advances, focussing on critical methodological developments that open up new research opportunities and challenges.

Reconstruction of network structure from multivariate time series of coupled dynamical systems is an important problem in multiple fields of science. This problem is ill-posed for large networks leading to the reconstruction of false network structures. We put forward the ergodic basis pursuit method that uses the network dynamics' statistical properties to ensure exact reconstruction of sparse networks when a minimum length of time series is attained. We show that this minimum length of time series scales quadratically with the node degree being probed and $\log$ of the system size. Our approach is robust against noise and allows us to treat the noise level as a parameter. We illustrate its applicability in experimental multivariate time series from optoelectronic networks.

We discuss our application of a method for reconstructing directed networks from dynamics [Ching and Tam, Phys. Rev. E 95, 010301(R) (2017)] to reveal the directed links and synaptic weights of cortical neuronal cultures from voltage measurements recorded by a multielectrode array. The effective connectivity so obtained reproduces several features of cortical regions in rats and monkeys and has similar network properties as the synaptic network of the nematode Caenorhabditis elegans, whose entire nervous system has been mapped out. The effective connectivity captures different information from the commonly studied functional connectivity, estimated using statistical correlation between spiking activities. The average synaptic strengths of excitatory incoming and outgoing links are found to increase with the spiking activity in the estimated effective connectivity but not in the functional connectivity estimated using the same sets of voltage measurements.

It has been postulated that information processing in the brain is based on precise temporal correlation of neural activity across populations of neurons. In a recent study we found spatio- temporal spike patterns in experimental recordings (10x10 Utah array) from monkey motor cortex using the SPADE method [1,2]. In 20 sessions each of 15 min recordings from monkeys performing a reach-to-grasp task, we found spatio-temporal spike patterns (STPs) with a precision of 5ms. These STPs were specific to the behavioral epoch (1-2 STPs) the monkey was in. The patterns typically contained 2-6 spikes from different neurons, and their duration varied from 5-60ms. These patterns were not clustered in cortical space, but distributed in space. Particularly surprising was, that individual neurons participate in different STPs occuring in different behavioral epochs [3]. This also means that neurons are part in different cell assemblies. In a further study, we were wondering if those STPs could be explained by a synfire chain (SFC) like model. An SFC is composed of groups of neurons connected in feed-forward manner from one group to the next with high convergence and divergence. The activity in such an SFC is, when activated e.g. by a current pulse to the first group, synchronous spiking activity within the neurons of a group, and this synchronous activity is propagating through the network. When a few neurons from different groups are assumed to be recorded from such an SFC we would find an STP repeating across trials. We consider the statistics of STPs found in the above mentioned experiment and compare them to recordings from a simulated network. The properties of the data are summarized in histograms showing 1) the pattern sizes, i.e. the number of neurons involved in the patterns, 2) the number of patterns a single neuron is involved in, 3) the durations of the patterns, and 4) the spatial distances of the patterns across the array. For the simulations, we embed SFC(s) in an anatomical model of the respective layer of the motor cortex, defined by its height and the density of the neurons. Model parameters are length of the SFC, number of neurons per group and the spatial embedding parameters the radius of the neurons per group are in and the distance between groups. Given the size and reach of the Utah electrodes, we derive the probability of recording neurons from the SFC network. An SFC is considered detected, if at least two neurons from two different groups are recorded from. We find, that depending on the SFC parameters, an embedded SFC is detectable with high probability, despite the massive subsampling by the Utah electrode. Furthermore, to achieve multiple membership of a neuron if different patterns, we have to assume the embedding of multiple SFCs. The fitting of the model to the pattern data constrains the spatial SFC parameters: the chains have to be broadly distributed in space and contain many neurons per group to match the experimental results [4]. 1. Stella A., Quaglio P., Torre E., Grün S. (2019) 3d-SPADE: Significance evaluation of spatio-temporal patterns of various temporal extents. Biosystems, 185: 104022. DOI: 10.1016/j.biosystems.2019.104022 2. Stella A., Bouss P., Palm G., Grün S. (2022) Comparing surrogates to evaluate precisely timed higher-order spike correlations. ENeuro 9 (3), DOI: https://doi.org/10.1523/ENEURO.0505-21.2022 3. Grün S, Li J, McNaughton B, Petersen C, McCormick D, Robson D, Buzaki G, Harris K, Sejnowski T, Mrsic-Flogel T, Linden H, Roland PE (2022) Emerging principles of spacetime in brains: Meeting report on spatial neurodynamics. Neuron 110(12), https://doi.org/10.1016/j.neuron.2022.05.018 4. Kleinjohann A, Berling D, Stella A, Tetzlaff T, Grün S. Detectability of spatiotemporal activity patterns arising from cortical feedforward networks. In preparation.

We propose a novel robust decentralized graph clustering algorithm that is provably equivalent to the popular spectral clustering approach. Our proposed method uses the existing wave equation clustering algorithm that is based on propagating waves through the graph. However, instead of using a fast Fourier transform (FFT) computation at every node, our proposed approach exploits the Koopman operator framework. Specifically, we show that propagating waves in the graph followed by a local dynamic mode decomposition (DMD) computation at every node is capable of retrieving the eigenvalues and the local eigenvector components of the graph Laplacian, thereby providing local cluster assignments for all nodes. We demonstrate that the DMD computation is more robust than the existing FFT based approach and requires 20 times fewer steps of the wave equation to accurately recover the clustering information and reduces the relative error by orders of magnitude. We demonstrate the decentralized approach on a range of graph clustering problems.

A novel methodology for inferring equations of motion of high-dimensional systems from time series data is introduced. Pairs of patterns, defined in the space of candidate functions and in the state space of the system, are identified and linked in a predictive manner. Numerical integration schemes are embedded in the method, allowing to extract differential equations while working on temporal data increments rather than numerical time derivatives which are very sensitive to noise. Examination of the pattern amplitudes allows for assessing the predictive contribution of certain terms, a systematic model pruning and the construction of a parsimonious or sparse model. The technique can be modified to deal with measurement noise and, under certain assumptions, with only partial observation of the system. The method is exemplified on the Lorenz 1996 system and a chain of oscillators.

To be provided soon.