Abstract
This paper introduces a class of stochastic models of interacting neurons with emergent dynamics similar to those seen in local cortical populations. Rigorous results on existence and uniqueness of nonequilibrium steady states are proved. These network models are then compared to very simple reduced models driven by the same mean excitatory and inhibitory currents. Discrepancies in firing rates between network and reduced models are investigated and explained by correlations in spiking, or partial synchronization, working in concert with “nonlinearities” in the time evolution of membrane potentials. The use of simple random walks and their first passage times to simulate fluctuations in neuronal membrane potentials and interspike times is also considered.
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Notes
We would like to clarify our use of the word “synchrony”, as it is used in many different ways in the neuroscience literature. In this paper, we have used it as a descriptive rather than technical term. By “synchronous behavior”, we refer to the more-than-coincidental near-simultaneous spiking of a substantial group of neurons, occurring repeatedly over time possibly with different groups of neurons participating in each event. Stronger synchrony refers to either a larger group of neurons participating in each spiking event, or each event concentrated in a smaller time window. We are aware that mathematicians sometimes use the word “synchrony” to mean perfectly-timed, full-population spikes; we do not mean that. (Brunel 2000) introduced the terms “synchronous” and “asynchronous states” to correspond to the population firing rate being oscillatory or constant in time; we also do not mean that. The models we consider are never synchronous in the sense of population spikes, and never asynchronous in the sense of Brunel, nor do we regard systems with small temporal oscillations in their population firing rate as being “synchronous” behavior.
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LC was supported by a grant from the Swartz Foundation.
LSY was supported in part by NSF Grant DMS-1363161.
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Li, Y., Chariker, L. & Young, LS. How well do reduced models capture the dynamics in models of interacting neurons?. J. Math. Biol. 78, 83–115 (2019). https://doi.org/10.1007/s00285-018-1268-0
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DOI: https://doi.org/10.1007/s00285-018-1268-0