CC BY
19
Section COMPUTATIONAL NEUROSCIENCE
Model of Neuronal Activity in Cultural Network with Energy Feedback
F.D.Iudin1*, D.I. Iudin2, A.N. Gorban2, T.A. Tyukina2,1.V. Mukhina1, V.B. Kazantsev1, and I.Yu. Tyukin3
1 Nizhny Novgorod State University, Nizhny Novgorod, Russia;
2 Institute of Applied Physics of RAS, Nizhny Novgorod, Russia;
3 Dept of Mathematics, University of Leicester, Leicester, United Kingdom. * Presenting e-mail: fd.iudin@gmail.com
Living networks in dissociated neuronal cultures are widely known for their ability to generate spatiotemporal activity patterns that satisfy the power scaling law and thereby exemplify self-organized criticality in living systems. Here we propose a simple network model with energy feedback regulating of the strength of local neural connectivity. Such regulatory mechanism results in the overall model behavior that can be characterized as balancing on the edge of the network percolation transition. Network activity in this state shows population bursts satisfying the scaling avalanche conditions. This network state is self-sustainable and represents a kind of energetic balance between global structural network-wide processes and spontaneous activity of individual elements.
Methods
In this work we further contribute to the idea that several features of SOC-like behavior (e.g. the neuonal avalanches, periodic and chaotic spiking) observed in live neuronal cultures and networks can be explained by local connectivity patterns, expressed by probabilities of connections between cells, neuronal activation dynamics, and by an additional regulatory variable that can be viewed as a generalized an energy supply. In our approach the network topology formation is assessed within the framework of percolation thresholds. Using this framework we establish critical connectivity parameters and employ them in dynamical models of neural activity. At first a simple percolation-based geometric model describing the evolution of cells' connectivity is presented. The model allows to accommodate biologically relevant features such as axons and dendrites; it also enables to replicate directional connectivity that is inherent to living systems including neuronal cultures. The model analysis reveals that sharp changes in the overall clustering and connectivity of the evolving network in both directed and undirected settings is determined by a single parameter describing average connection density in the network. After that we present a mean-field approximation to neuronal activity in cultures and show that if the model is supplied with energy-dependent activation then the connectivity parameter may be used to describe periodic spiking, irregular dynamics, and population bursts. And at last, we provide results of large-scale simulation of evolving network of neuron-like agents of which the activation probability depends on their current energy level.
Results
Our geometric model reveals that networks with coordination numbers exceeding these critical values are likely to form spanning clusters that are capable of connecting nearly all elements in the system. The mean-field dynamic model contains two additional variables: one is the maximal probability of neuronal activation in response to incoming spike, and the other is an exogenous "resource" variable determining if a neuron has enough energy to elicit a spike. The mean-field bursting dynamics resembles that of the population bursts observed in live neuronal evolving cultures. An important factor in successful replication of this behavior was the energy variable coupled with the energy-dependent activation probability. The mean-field model, however, lacks spatial variability and as such is only a rough approximation of activity propagation in neuronal cultures. In our multi-agent model emerging population bursts appear to be synchronized with the peaks of the energy function, because in network models the connections change according to rate dependent synaptic plasticity. Moreover, the same behavior can be observed in the dynamics of mean-field approximation. Hence, the exogenous energy variable introduced here from mere phenomenological considerations might constitute a macroscopical model of how the neuron's ability to transmit spikes influence the network's dynamics. Finally, since in our simulations network connectivity was always kept above the corresponding percolation thresholds neuronal avalanches where observed. The avalanches, however, appeared only when the energy level was sufficiently high.
Conclusion
In summary, we have proposed a network model explaining burst generation in living networks. A distinct feature of our model is presence of a dynamic exogenous energy variable and neuronal activation probability that is made dependent on the energy. We showed that introduction of these modifications already enables to explain evolution
OM&P
OM&P
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
