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Daie K, Fontolan L, Druckmann S, Svoboda K. Feedforward amplification in recurrent networks underlies paradoxical neural coding. BIORXIV : THE PREPRINT SERVER FOR BIOLOGY 2023:2023.08.04.552026. [PMID: 37577599 PMCID: PMC10418196 DOI: 10.1101/2023.08.04.552026] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 08/15/2023]
Abstract
The activity of single neurons encodes behavioral variables, such as sensory stimuli (Hubel & Wiesel 1959) and behavioral choice (Britten et al. 1992; Guo et al. 2014), but their influence on behavior is often mysterious. We estimated the influence of a unit of neural activity on behavioral choice from recordings in anterior lateral motor cortex (ALM) in mice performing a memory-guided movement task (H. K. Inagaki et al. 2018). Choice selectivity grew as it flowed through a sequence of directions in activity space. Early directions carried little selectivity but were predicted to have a large behavioral influence, while late directions carried large selectivity and little behavioral influence. Consequently, estimated behavioral influence was only weakly correlated with choice selectivity; a large proportion of neurons selective for one choice were predicted to influence choice in the opposite direction. These results were consistent with models in which recurrent circuits produce feedforward amplification (Goldman 2009; Ganguli et al. 2008; Murphy & Miller 2009) so that small amplitude signals along early directions are amplified to produce low-dimensional choice selectivity along the late directions, and behavior. Targeted photostimulation experiments (Daie et al. 2021b) revealed that activity along the early directions triggered sequential activity along the later directions and caused predictable behavioral biases. These results demonstrate the existence of an amplifying feedforward dynamical motif in the motor cortex, explain paradoxical responses to perturbation experiments (Chettih & Harvey 2019; Daie et al. 2021b; Russell et al. 2019), and reveal behavioral relevance of small amplitude neural dynamics.
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Duan C, Nishikawa T, Eroglu D, Motter AE. Network structural origin of instabilities in large complex systems. SCIENCE ADVANCES 2022; 8:eabm8310. [PMID: 35857524 PMCID: PMC9286506 DOI: 10.1126/sciadv.abm8310] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 10/14/2021] [Accepted: 06/02/2022] [Indexed: 06/15/2023]
Abstract
A central issue in the study of large complex network systems, such as power grids, financial networks, and ecological systems, is to understand their response to dynamical perturbations. Recent studies recognize that many real networks show nonnormality and that nonnormality can give rise to reactivity-the capacity of a linearly stable system to amplify its response to perturbations, oftentimes exciting nonlinear instabilities. Here, we identify network structural properties underlying the pervasiveness of nonnormality and reactivity in real directed networks, which we establish using the most extensive dataset of such networks studied in this context to date. The identified properties are imbalances between incoming and outgoing network links and paths at each node. On the basis of this characterization, we develop a theory that quantitatively predicts nonnormality and reactivity and explains the observed pervasiveness. We suggest that these results can be used to design, upgrade, control, and manage networks to avoid or promote network instabilities.
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Affiliation(s)
- Chao Duan
- School of Electrical Engineering, Xi’an Jiaotong University, Xi’an, 710049, China
- Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA
| | - Takashi Nishikawa
- Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA
- Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL 60208, USA
| | - Deniz Eroglu
- Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA
- Department of Molecular Biology and Genetics, Kadir Has University, 34083 Istanbul, Turkey
| | - Adilson E. Motter
- Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA
- Northwestern Institute on Complex Systems, Northwestern University, Evanston, IL 60208, USA
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Rodgers N, Tiňo P, Johnson S. Network hierarchy and pattern recovery in directed sparse Hopfield networks. Phys Rev E 2022; 105:064304. [PMID: 35854620 DOI: 10.1103/physreve.105.064304] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/01/2022] [Accepted: 05/22/2022] [Indexed: 06/15/2023]
Abstract
Many real-world networks are directed, sparse, and hierarchical, with a mixture of feedforward and feedback connections with respect to the hierarchy. Moreover, a small number of master nodes are often able to drive the whole system. We study the dynamics of pattern presentation and recovery on sparse, directed, Hopfield-like neural networks using trophic analysis to characterize their hierarchical structure. This is a recent method which quantifies the local position of each node in a hierarchy (trophic level) as well as the global directionality of the network (trophic coherence). We show that even in a recurrent network, the state of the system can be controlled by a small subset of neurons which can be identified by their low trophic levels. We also find that performance at the pattern recovery task can be significantly improved by tuning the trophic coherence and other topological properties of the network. This may explain the relatively sparse and coherent structures observed in the animal brain and provide insights for improving the architectures of artificial neural networks. Moreover, we expect that the principles we demonstrate here, through numerical analysis, will be relevant for a broad class of system whose underlying network structure is directed and sparse, such as biological, social, or financial networks.
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Affiliation(s)
- Niall Rodgers
- School of Mathematics, University of Birmingham, Birmingham B15 2TT, United Kingdom and Topological Design Centre for Doctoral Training, University of Birmingham, Birmingham B15 2TT, United Kingdom
| | - Peter Tiňo
- School of Computer Science, University of Birmingham, Birmingham B15 2TT, United Kingdom
| | - Samuel Johnson
- School of Mathematics, University of Birmingham, Birmingham B15 2TT, United Kingdom and The Alan Turing Institute, British Library, 96 Euston Road, London NW1 2DB, United Kingdom
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Makris KG. Transient growth and dissipative exceptional points. Phys Rev E 2021; 104:054218. [PMID: 34942815 DOI: 10.1103/physreve.104.054218] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/27/2021] [Accepted: 09/19/2021] [Indexed: 11/07/2022]
Abstract
In the context of non-Hermitian photonics, we study the physics of transient growth in coupled waveguide systems that exhibit higher-order exceptional points. We demonstrate the counterintuitive effect of transient growth despite the decaying spectrum, which is a direct consequence of the underlying modal nonorthogonality. Eigenvalue analysis fails to capture the power dynamics and thus we have to rely on methods of nonmodal stability theory, namely singular value decomposition and pseudospectra. The relation between the order of the exceptional point and transient growth is also examined. Our work provides a general methodology that can be applied to any non-Hermitian system that contains complex elements with more loss than gain, and exploits the boundaries of transient amplification in dissipative environments.
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Affiliation(s)
- K G Makris
- ITCP-Physics Department, University of Crete, 71003 Heraklion, Greece and Institute of Electronic Structure and Laser, FORTH, 71110 Heraklion, Greece
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Abstract
Out of equilibrium, a lack of reciprocity is the rule rather than the exception. Non-reciprocity occurs, for instance, in active matter1-6, non-equilibrium systems7-9, networks of neurons10,11, social groups with conformist and contrarian members12, directional interface growth phenomena13-15 and metamaterials16-20. Although wave propagation in non-reciprocal media has recently been closely studied1,16-20, less is known about the consequences of non-reciprocity on the collective behaviour of many-body systems. Here we show that non-reciprocity leads to time-dependent phases in which spontaneously broken continuous symmetries are dynamically restored. We illustrate this mechanism with simple robotic demonstrations. The resulting phase transitions are controlled by spectral singularities called exceptional points21. We describe the emergence of these phases using insights from bifurcation theory22,23 and non-Hermitian quantum mechanics24,25. Our approach captures non-reciprocal generalizations of three archetypal classes of self-organization out of equilibrium: synchronization, flocking and pattern formation. Collective phenomena in these systems range from active time-(quasi)crystals to exceptional-point-enforced pattern formation and hysteresis. Our work lays the foundation for a general theory of critical phenomena in systems whose dynamics is not governed by an optimization principle.
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Synchronization Dynamics in Non-Normal Networks: The Trade-Off for Optimality. ENTROPY 2020; 23:e23010036. [PMID: 33383735 PMCID: PMC7823721 DOI: 10.3390/e23010036] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 12/05/2020] [Revised: 12/22/2020] [Accepted: 12/24/2020] [Indexed: 11/16/2022]
Abstract
Synchronization is an important behavior that characterizes many natural and human made systems that are composed by several interacting units. It can be found in a broad spectrum of applications, ranging from neuroscience to power-grids, to mention a few. Such systems synchronize because of the complex set of coupling they exhibit, with the latter being modeled by complex networks. The dynamical behavior of the system and the topology of the underlying network are strongly intertwined, raising the question of the optimal architecture that makes synchronization robust. The Master Stability Function (MSF) has been proposed and extensively studied as a generic framework for tackling synchronization problems. Using this method, it has been shown that, for a class of models, synchronization in strongly directed networks is robust to external perturbations. Recent findings indicate that many real-world networks are strongly directed, being potential candidates for optimal synchronization. Moreover, many empirical networks are also strongly non-normal. Inspired by this latter fact in this work, we address the role of the non-normality in the synchronization dynamics by pointing out that standard techniques, such as the MSF, may fail to predict the stability of synchronized states. We demonstrate that, due to a transient growth that is induced by the structure’s non-normality, the system might lose synchronization, contrary to the spectral prediction. These results lead to a trade-off between non-normality and directedness that should be properly considered when designing an optimal network, enhancing the robustness of synchronization.
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Bernal Jaquez R, Alarcón Ramos LA, Schaum A. Spreading Control in Two-Layer Multiplex Networks. ENTROPY (BASEL, SWITZERLAND) 2020; 22:E1157. [PMID: 33286926 PMCID: PMC7597322 DOI: 10.3390/e22101157] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 09/03/2020] [Revised: 09/30/2020] [Accepted: 10/08/2020] [Indexed: 01/18/2023]
Abstract
The problem of controlling a spreading process in a two-layer multiplex networks in such a way that the extinction state becomes a global attractor is addressed. The problem is formulated in terms of a Markov-chain based susceptible-infected-susceptible (SIS) dynamics in a complex multilayer network. The stabilization of the extinction state for the nonlinear discrete-time model by means of appropriate adaptation of system parameters like transition rates within layers and between layers is analyzed using a dominant linear dynamics yielding global stability results. An answer is provided for the central question about the essential changes in the step from a single to a multilayer network with respect to stability criteria and the number of nodes that need to be controlled. The results derived rigorously using mathematical analysis are verified using statical evaluations about the number of nodes to be controlled and by simulation studies that illustrate the stability property of the multilayer network induced by appropriate control action.
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Affiliation(s)
- Roberto Bernal Jaquez
- Department of Applied Mathematics and Systems, Universidad Autónoma Metropolitana, Cuajimalpa, Mexico-City 05348, Mexico;
| | - Luis Angel Alarcón Ramos
- Postgraduate in Natural Sciences and Engineering, Universidad Autónoma Metropolitana, Cuajimalpa, Mexico-City 05348, Mexico
| | - Alexander Schaum
- Chair of Automatic Control, Kiel-University, 24143 Kiel, Germany;
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