Hierarchy of brain oscillations emerge from recurrent error correction

Neuronal processing in the brain occurs rhythmically across a set of discrete frequency bands, spanning almost two orders of magnitude. The origin of these rhythms remains a matter of debate, as well as why activity appears to be organized in canonical bands. Here we demonstrate that the relative distribution of frequency bands emerge from the dynamics of recurrent neural networks (RNNs) performing error correction. These networks achieve best performance when processing pulsed inputs with noise levels proportional to those observed in cortical networks. In this optimal regime, the performance timescale is T₀ ≈ 2.14 ∗ τ , where τ denotes the integration time constant for network nodes. A minimal timescale T₀ can be obtained for recurrent networks composed of individual neurons. Longer timescales T_n are sequentially derived when the outputs of individual RNNs become the nodes of a higher order recurrent network of their own, where T_n = T₀ ∗ (T₀ /τ)^{(n - 1)}. We show that this pattern of timescales reproduces the canonical oscillatory bands seen in neural data. The intrinsic timescale T₀ can be reduced by increasing the gain of individual nodes. This could allow neuromodulatory or attentional gain mechanisms to modulate the processing speed of salient signals. Successful error correction can be achieved in small networks with several neurons, with no substantial benefit of using larger networks. These results describe a mechanism through which the empirically observed discrete set of frequency bands can emerge through hierarchically organized RNNs.