Slow timescales generically emerge from divisive normalization

The brain orchestrates behaviors that span milliseconds to years. While a single long timescale can emerge in recurrent linear models without significant fine-tuning of neural interactions, it remains unclear how a spectrum of slow timescales emerges from faster neural activity without fine-tuning. We find that many slow timescales can emerge naturally from large recurrent neural networks implementing divisive normalization, where each neuron's activity is modulated by the activity of similarly tuned neurons in response to external stimuli. It has already been reported that divisive normalization stabilizes recurrent networks that otherwise would be unstable. We demonstrate that marginally normalized recurrent networks, or networks where divisive normalization is weak, have slow dynamics and functional connectivity that scales subextensively with system size. Specifically, using a recurrent model for divisive normalization, we show that the decay rates of marginally normalized networks follow the Kesten McKay distribution for graphs where each neuron has d connections to other neurons. In agreement with the properties of these d-regular graphs, the marginally normalized neurons' slowest decay rates slow down subextensively with system size as N^{-2/3}. Furthermore, the network's functional connectivity d increases subextensively with system size as N^{2/3}. Our results suggest that weak engagement of divisive normalization generically gives rise to a wide range of long timescales. This key prediction of our theory can be tested by estimating per-neuron decay rates from simultaneous population recordings and assessing whether their empirical distribution follows the Kesten–McKay form. Such an experimental confirmation of our theory would elucidate the interplay of slow timescales, stability, and functional connectivity in neural systems, given that divisive normalization is ubiquitous in neural systems across species.