A minimal neural network at the edge of chaos

Experimental results show that biological neural networks display spontaneous transitions between ordered and chaotic activity. However, in vitro measurements of the electrical signals fired by single neurons show a regular, reproducible dynamics, implying that the chaotic behavior is an emerging property of the network. A frequently adopted hypothesis is that chaos emerges as a consequence of a fine tuned balance between the excitatory and inhibitory neural connections in large networks with high connectivity. Here we argue that while this hypothesis is sufficient for explaining the emergence of complex behavior, it may not be necessary for explaining the observed spontaneous order-chaos transitions. More exactly, we show that simple neural oscillators can exhibit order-chaos transitions when forced by periodic signals generated by other neurons. In this model, the simplest such neural network consists of only three neurons connected in a master-slave architecture (one master and two slave neurons). We also show, both theoretically and using an electronic circuit model, that such a network can switch its dynamics between the regular and chaotic regimes by simply changing the firing signal frequency of the master (controlling) neuron. These results show that an increased network complexity is not necessary for explaining the spontaneous order-chaos transitions frequently observed in biological neural networks.