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Section COMPUTATIONAL NEUROSCIENCE
of cultures from resting state to population bursts, at least in the mean-field approximation. Large-scale multi-agent simulations enabled to demonstrate that these additional variables representing very basic physical mechanisms, including energy feedback are capable of stirring the network's dynamical state to the edge of percolation transition.
With respect to dynamics of signals in the network, we have shown that population bursts naturally arise in the network's state corresponding to the energy homeostasis. Statistical properties of the bursts, as was observed in many experimental and theoretical studies, inherit essential features of SOC-statistics: e.g. presence of multiscale excitations. Given that systems operating in the percolation state are exhibit increased sensitivity to external perturbations, we can conclude that networks operating in this regime may offer increased responsiveness to incoming stimulation.
Finally, we showed that a network in which the evolution of connectivity is balanced by the generalized energy consumption eventually arrives at the dynamical state characterized by extremely robust and persistent bursting.
Acknowledgements
The research was supported by the Russian Science Foundation (Agreement 14-19-01381) and by the Russian Foundation for Basic Research (project 15-38-20178).
References
1. F.D. Iudin, D.I. Iudin, V.B. Kazantsev. Percolation treshold in active neural networks with adaptive geometry, JETP Lett., 101 (4), 2015, 271-275.
2. A.S. Pimashkin, I.A. Kastalskiy, A.Yu. Simonov, E.A. Koryagina, S.A. Korotchenko, I.V. Mukhina , V.B. Kazantsev. Spiking signatures of spontaneous activity bursts in hippocampal cultures. Frontiers in Computational Neuroscience, 5(46), 2011, 1-12.
3. D.I. Iudin, Ya.D. Sergeyev, M. Hayakawa. Interpretation of percolation in terms of infinity computations. Applied Mathematics and Computation. 218(16), 2012, 8099-8111.
4. A.N. Gorban, T.A. Tyukina, E.V. Smirnova, L.I. Pokidysheva. Evolution of adaptation mechanisms: adaptation energy, stress, and oscillating death, J. Theor. Biol., in press, 2015.
multtstability and coherent dynamics in directed networks of
Heterogeneous Neural Oscillators with Modular Network Topologies
I.Y. Tyukin1,2*, E. Steur3, A.N. Gorban1, N. Jarman1,4, H. Nijmeijer3, C. van Leeuwen4
1 University of Leicester, Leicester, United Kingdom;
2 Saint-Petersburg State Electrotechnical University, Saint-Petersburg, Russia;
3 Eindhoven University of Technology, Eindhoven, The Netherlands;
4 University of Leuven, Leuven, Belgium. * Presenting e-mail: i.tyukin@le.ac.uk
Understanding the dynamics of interconnected systems of nonlinear ordinary differential equations is arguably amongst the oldest and inspiring problems. Objects of this type occur in a broad range of fields of engineering and science [1]. Significant progress has been made in this area with regards to general laws governing the emergence of various synchronous states, see e.g. [2] and references therein; and the presence of intricate dependencies between network topologies, properties of individual nodes and dynamics in networks have now been elucidated by many authors [3], [4], [5], [6], [7], [8], [9]. Despite this progress, however, a few fundamental questions remain, including the question, how a specific configuration of network topology and weights may affect the overall behavior of the network.
It has been shown recently in [10], [11] that "closing" a chain of identical nonlinear oscillators with directed coupling by adding a directed feedback from the last element in the chain to the first dramatically affects the dynamics of the system. In the chain one only finds a single coherent state, the full synchronization. Closing the chain results in the creation of a new system, in which multiple coherent states may coexist: rotating wave solutions of various modes and a fully synchronous state. The observed abrupt change in dynamics has been attributed to the behavior of the spectrum of the network Laplacian matrix. Furthermore, it has also been shown in [10], [11] that rotating wave solutions prevail in long directed cycles.
In this work we develop and generalize these results in the following two directions. First, instead of directed chains we consider networks with modular structure. Such networks comprise of diffusively and undirectly coupled groups of nodes (modules). These groups are linked by directed connections forming a directed cycle. We show that, remarkably, the spectrum of the network Laplacian for such modular structures is closely related to that of individual isolated modules and the corresponding ring or cycle. Similar to our previous work [11] we hypothesise that rotating wave solutions are likely in such networks. In addition, rotating wave solutions are expected to occur more frequently in the directed cycle of modules than
Section COMPUTATIONAL NEUROSCIENCE
in the directed cycle of simple oscillators. Numerical simulations confirm this hypothesis. Second, in addition to nodes with identical dynamics, we numerically investigate the case in which individual oscillators differ; their parameters are randomly sampled from a distribution centered at fixed nominal values. We observe that, provided that coupling within individual modules is strong enough, solutions resembling rotating waves emerge in this system, too. The latter regime co-exist with nearly synchronous state giving rise to coupling strength-modulated multi-stability and coherence in such systems.
Acknowledgements
The first author is thankful to the RFBR (grant 15-38-20178) for partial support of this work. References
1. W. Strogate. Sync: How Order Emerges From Chaos In the Universe, Nature, and Daily Life. Hyperion, 2003.
2. A. Pikovsky, M. Rosenblum, and J. Kurths. Synchroniozation A universal concept in nonlinear sciences. Cambridge University Press, 2001.
3. M. Golubitsky, I. Stewart, I., and A. Torok SIAM Journal on Applied Dynamical Systems, 2005, 4(1), 78-100.
4. A.Y. Pogromsky. Int. J. of Bifurc. and Chaos, 1998, 8(2), 295-319.
5. L. Scardovi, M. Arcak, and E. Sontag. IEEETransactions on Automatic Control, 2010, 55, 1367-1379.
6. A. Pogromsky, G. Santoboni, and H. Nijmeijer. Physica D : Nonlinear Phenomena, 2002, 172(1-4), 65-87.
7. I. Belykh, V. Belykh, and M. Hasler. Physical Review E, 2000, 62(5), 6332-6345.
8. I. Belykh, V. Belykh, and M. Hasler. Physica D: Nonlinear Phenomena, 2004, 195(1-2), 159-187.
9. V. Chandrasekar, J. Sheeba, B. Subash, M. Lakshmanan, and J. Kurths. Physica D: Nonlinear Phenomena, 2014, 267, 36-48.
10. A. Gorban, N. Jarman, E. Steur, H. Nijmeijer, C. van Leeuwen, and I. Tyukin Proceedings of the 4-th IFAC Conference on Control of Chaotic Systems, 2015.
11. A. Gorban, N. Jarman, E. Steur, C. van Leeuwen, and I. Tyukin. Mathematical Modelling of Natural Phenomena, 10, 212-231.
Cocaine Addiction as a Homeostahc Reinforcement Learning Disorder
Boris S. Gutkin1,2*
1 Group for Neural Theory, LNC INSERM U960, Ecole Normale Superieure, Paris, France;
2 Center for Cognition and Decision Making, NRU Higher School of Economics, Moscow, Russia. * Presenting e-mail: boris.gutkin@gmail.com
Background
Current addiction theories diverge and depict cocaine addiction as a disorder of either reinforcement learning or hedonic homeostatic regulation. The learning-based theories, in particular, suggests that addiction results from transition from a voluntary and goal-directed, to a habitual decision process. Underlying all of these computational theories is the central role of the dopaminergic system signaling reward information. Under cocaine, as under other addictive drugs, this signaling is pathologically influenced by the pharmacological action of the drug.
Methods
Here we propose a new theory of addiction that integrates the brain reinforcement learning and hedonic homeostatic regulation systems. As opposed to habit-based theories, we postulate that in addicts, a goal-directed planning system remains in charge of fulfilling drug-related homeostatic needs of the organism, while drugs gradually modify the need structure, as well as associative learning mechanisms. Formal mathematical modeling and simulation, as well as cocaine self-administration paradigm in rats are combined to test key predictions of our theory.
Results
Simulations show that our new theory accounts for key behavioral and neurobiological features of addiction, most notably, escalation of cocaine use, drug-primed craving and relapse, individual differences underlying susceptibility to addiction, and dopamine D2-receptor down-regulation in addicts. The theory also generates unique predictions about cocaine self-administration behavior in rats that are confirmed by new experimental results.
Conclusion
We show that our integrative theory explains many behavioral and neurobiological aspects of cocaine addiction that
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