Research Article: Simple model of complex dynamics of activity patterns in developing networks of neuronal cultures

Date Published: June 27, 2019

Publisher: Public Library of Science

Author(s): Ivan Y. Tyukin, Dmitriy Iudin, Feodor Iudin, Tatiana Tyukina, Victor Kazantsev, Irina Mukhina, Alexander N. Gorban, Gennady Cymbalyuk.


Living neuronal networks in dissociated neuronal cultures are widely known for their ability to generate highly robust spatiotemporal activity patterns in various experimental conditions. Such patterns are often treated as neuronal avalanches that satisfy the power scaling law and thereby exemplify self-organized criticality in living systems. A crucial question is how these patterns can be explained and modeled in a way that is biologically meaningful, mathematically tractable and yet broad enough to account for neuronal heterogeneity and complexity. Here we derive and analyse a simple network model that may constitute a response to this question. Our derivations are based on few basic phenomenological observations concerning the input-output behavior of an isolated neuron. A distinctive feature of the model is that at the simplest level of description it comprises of only two variables, the network activity variable and an exogenous variable corresponding to energy needed to sustain the activity, and few parameters such as network connectivity and efficacy of signal transmission. The efficacy of signal transmission is modulated by the phenomenological energy variable. Strikingly, this simple model is already capable of explaining emergence of network spikes and bursts in developing neuronal cultures. The model behavior and predictions are consistent with published experimental evidence on cultured neurons. At the larger, cellular automata scale, introduction of the energy-dependent 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 energetic balance between global network-wide processes and spontaneous activity of individual elements.

Partial Text

Exploiting physics’ concepts for dealing with problems in life sciences is a widely recognized and successful strategy for developing systematic and lawful understanding of complex phenomena observed in empirical data. One of the most striking and fashionable illustrations facilitating potential and power of this approach is the well-known example of using the concept of self-organized criticality (SOC)—the ability of systems to selftune to the critical state—for explaining a number of puzzling effects in biological systems. Initially proposed as a model for explaining how an abstract system can remain at a critical state in presence of perturbations [1, 2], the concept is now broadly used for describing biological neural networks (see e.g. [3, 4]). It was shown that adaptively evolving networks, i.e., networks combining structural evolution of the network topology with dynamics in the network nodes [5], can exhibit highly robust global SOC-like behavior maintained by simple local network adjustment rules.

In summary, we have proposed a simple network model explaining burst generation in living culture 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, like in general models of physiological adaptation [28]. We showed that introduction of these modifications already enables to explain evolution of cultures from resting state to population bursts, at least in the mean-field approximation. In accordance to the model, emergence of bursts and spikes is regulated by just few parameters that correspond to network connectivity and efficacy of synaptic transmission. We also note that our energy-based model is complementary to more traditional connectivity-focused approaches [23].

No animals or human subjects were used or employed as a part of research presented in this work.