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

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.