## Section: New Results

### Neural Networks as dynamical systems

#### Metastable Resting State Brain Dynamics

Participants : Peter Beim Graben [Brandenburg University of Technology Cottbus, Germany] , Antonio Jimenez-Marin [Computational Neuroimaging Lab, BioCruces-Bizkaia Health Research Institute, Spain] , Ibai Diez [Harvard Medical School, Massachusetts General Hospital, Boston, MA, USA] , Jesus M Cortes [Computational Neuroimaging Lab, BioCruces-Bizkaia Health Research Institute, Spain] , Mathieu Desroches, Serafim Rodrigues [Ikerbasque & MCEN team, Basque Center for Applied Mathematics, Spain] .

Metastability refers to the fact that the state of a dynamical system spends a large amount of time in a restricted region of its available phase space before a transition takes place, bringing the system into another state from where it might recur into the previous one. Beim Graben and Hutt (2013) [74] suggested to use the recurrence plot (RP) technique introduced by Eckmann et al. (1987) [57] for the segmentation of system's trajectories into metastable states using recurrence grammars. Here, we apply this recurrence structure analysis (RSA) for the first time to resting-state brain dynamics obtained from functional magnetic resonance imaging (fMRI). Brain regions are defined according to the brain hierarchical atlas (BHA) developed by Diez et al. (2015) [56], and as a consequence, regions present high-connectivity in both structure (obtained from diffusion tensor imaging) and function (from the blood-level dependent-oxygenation–BOLD–signal). Remarkably, regions observed by Diez et al. were completely time-invariant. Here, in order to compare this static picture with the metastable systems dynamics obtained from the RSA segmentation, we determine the number of metastable states as a measure of complexity for all subjects and for region numbers varying from 3 to 100. We find RSA convergence toward an optimal segmentation of 40 metastable states for normalized BOLD signals, averaged over BHA modules. Next, we build a bistable dynamics at population level by pooling 30 subjects after Hausdorff clustering. In link with this finding, we reflect on the different modeling frameworks that can allow for such scenarios: heteroclinic dynamics, dynamics with riddled basins of attraction, multiple timescale dynamics. Finally, we characterize the metastable states both functionally and structurally, using templates for resting state networks (RSNs) and the automated anatomical labeling (AAL) atlas, respectively.

This work has been published in Frontiers in Computational Neuroscience and is available as [20].

#### Controlling seizure propagation in large-scale brain networks

Participants : Simona Olmi, Spase Petkoski [Institut de Neurosciences des Systèmes, Marseille] , Maxime Guye [CEMEREM, Hôpital de la Timone, Marseille] , Fabrice Bartolomei [Hôpital de la Timone, Marseille] , Viktor Jirsa [Institut de Neurosciences des Systèmes, Marseille] .

Information transmission in the human brain is a fundamentally dynamic network process. In partial epilepsy, this process is perturbed and highly synchronous seizures originate in a local network, the so-called epileptogenic zone (EZ), before recruiting other close or distant brain regions. We studied patient-specific brain network models of 15 drug-resistant epilepsy patients with implanted stereotactic electroencephalography (SEEG) electrodes. Each personalized brain model was derived from structural data of magnetic resonance imaging (MRI) and diffusion tensor weighted imaging (DTI), comprising 88 nodes equipped with region specific neural mass models capable of demonstrating a range of epileptiform discharges. Each patients virtual brain was further personalized through the integration of the clinically hypothesized EZ. Subsequent simulations and connectivity modulations were performed and uncovered a finite repertoire of seizure propagation patterns. Across patients, we found that (i) patient-specific network connectivity is predictive for the subsequent seizure propagation pattern; (ii) seizure propagation is characterized by a systematic sequence of brain states; (iii) propagation can be controlled by an optimal intervention on the connectivity matrix; (iv) the degree of invasiveness can be significantly reduced via the here proposed seizure control as compared to traditional resective surgery. To stop seizures, neurosurgeons typically resect the EZ completely. We showed that stability analysis of the network dynamics using graph theoretical metrics estimates reliably the spatiotemporal properties of seizure propagation. This suggests novel less invasive paradigms of surgical interventions to treat and manage partial epilepsy.

This work has been published in PLoS Computational Biology and is available as [29].

#### Chimera states in pulse coupled neural networks: the influence of dilution and noise

Participants : Simona Olmi, Alessandro Torcini [Institute of Complex Systems, Florence, Italy] .

We analyse the possible dynamical states emerging for two symmetrically pulse coupled populations of leaky integrate-and-fire neurons. In particular, we observe broken symmetry states in this set-up: namely, breathing chimeras, where one population is fully synchronized and the other is in a state of partial synchronization (PS) as well as generalized chimera states, where both populations are in PS, but with different levels of synchronization. Symmetric macroscopic states are also present, ranging from quasi-periodic motions, to collective chaos, from splay states to population anti-phase partial synchronization. We then investigate the influence disorder, random link removal or noise, on the dynamics of collective solutions in this model. As a result, we observe that broken symmetry chimera-like states, with both populations partially synchronized, persist up to 80 % of broken links and up to noise amplitudes 8 % of threshold-reset distance. Furthermore, the introduction of disorder on symmetric chaotic state has a constructive effect, namely to induce the emergence of chimera-like states at intermediate dilution or noise level.

This work has been published as a chapter in the book Nonlinear Dynamics in Computational Neuroscience (Springer, 2019) and is available as [35].

#### Enhancing power grid synchronization and stability through time delayed feedback control

Participants : Halgurd Taher, Simona Olmi, Eckehard Schöll [Technical University Berlin, Germany] .

We study the synchronization and stability of power grids within the Kuramoto phase oscillator model with inertia with a bimodal frequency distribution representing the generators and the loads. We identify critical nodes through solitary frequency deviations and Lyapunov vectors corresponding to unstable Lyapunov exponents. To cure dangerous deviations from synchronization we propose time-delayed feedback control, which is an efficient control concept in nonlinear dynamic systems. Different control strategies are tested and compared with respect to the minimum number of controlled nodes required to achieve synchronization and Lyapunov stability. As a proof of principle, this fast-acting control method is demonstrated using a model of the German power transmission grid.

This work has been published in Physical Review E and is available as [32].

#### Stability and control of power grids with diluted network topology

Participants : Liudmila Tumash [gipsa-lab, CNRS, Grenoble] , Simona Olmi, Eckehard Schöll [Technical University Berlin, Germany] .

In the present study we consider a random network of Kuramoto oscillators with inertia in order to mimic and investigate the dynamics emerging in high-voltage power grids. The corresponding natural frequencies are assumed to be bimodally Gaussian distributed, thus modeling the distribution of both power generators and consumers: for the stable operation of power systems these two quantities must be in balance. Since synchronization has to be ensured for a perfectly working power grid, we investigate the stability of the desired synchronized state. We solve this problem numerically for a population of N rotators regardless of the level of quenched disorder present in the topology. We obtain stable and unstable solutions for different initial phase conditions, and we propose how to control unstable solutions, for sufficiently large coupling strength, such that they are stabilized for any initial phase. Finally, we examine a random Erdös-Renyi network under the impact of white Gaussian noise, which is an essential ingredient for power grids in view of increasing renewable energy sources.

This work has been published in Chaos: An Interdisciplinary Journal of Nonlinear Science and is available as [33].

#### Modeling dopaminergic modulation of clustered gamma rhythms

Participants : Denis Zakharov [Center for Cognition and Decision Making, HSE, Moscow, Russia] , Martin Krupa [UCA, LJAD, Inria MathNeuro] , Boris Gutkin [Laboratoire de Neurosciences Cognitives, ENS, Paris] .

Gamma rhythm (20-100Hz) plays a key role in numerous cognitive tasks: working memory, sensory processing and in routing of information across neural circuits. In comparison with lower frequency oscillations in the brain, gamma-rhythm associated firing of the individual neurons is sparse and the activity is locally distributed in the cortex. Such “weak” gamma rhythm results from synchronous firing of pyramidal neurons in an interplay with the local inhibitory interneurons in a “pyramidal-interneuron gamma” or PING. Experimental evidence shows that individual pyramidal neurons during such oscillations tend to fire at rates below gamma, with the population showing clear gamma oscillations and synchrony. One possible way to describe such features is that this gamma oscillation is generated within local synchronous neuronal clusters. The number of such synchronous clusters defines the overall coherence of the rhythm and its spatial structure. The number of clusters in turn depends on the properties of the synaptic coupling and the intrinsic properties of the constituent neurons. We previously showed that a slow spike frequency adaptation current in the pyramidal neurons can effectively control cluster numbers. These slow adaptation currents are modulated by endogenous brain neuromodulators such as dopamine, whose level is in turn related to cognitive task requirements. Hence we postulate that dopaminergic modulation can effectively control the clustering of weak gamma and its coherence. In this paper we study how dopaminergic modulation of the network and cell properties impacts the cluster formation process in a PING network model.

This work has been accepted for publication in Communications in Nonlinear Science and Numerical Simulation and is available as [34].