Communications in Mathematical Sciences

Volume 17 (2019)

Number 5

Dedicated to the memory of Professor David Shen Ou Cai

Coarse-grained descriptions of oscillations in neuronal network models

Pages: 1437 – 1458



Jennifer Crodelle (Courant Institute of Mathematical Sciences, New York University, New York, N.Y., U.S.A.)

Katherine A. Newhall (Department of Mathematics, University of North Carolina, Chapel Hill, N.C., U.S.A.)

Pamela B. Pyzza (Mathematics and Computer Science Department, Ohio Wesleyan University, Delaware, Oh., U.S.A.)

Gregor Kovačič (Mathematical Sciences Department, Rensselaer Polytechnic Institute, Troy, New York, U.S.A.)


Synchronous neuronal network oscillations are a ubiquitous phenomenon with great complexity of manifestations. We focus on coupled point-neuron models with increasing complexity to explore the similarities and differences in the underlying network mechanisms producing synchronous oscillations. Using simulations and coarse-grained descriptions, we illuminate mechanisms or mathematical structures that may be responsible for three stages of synchronous oscillations in the presence of noise: subthreshold dynamics, initiating a firing event, and synchronous termination of the event.


neuronal network, integrate-and-fire, Hodgkin-Huxley, synchrony, oscillations, bifurcation, stochastic dynamics

2010 Mathematics Subject Classification

37G15, 37N25, 82C31, 92B25

Received 23 April 2019

Accepted 5 September 2019

Published 6 December 2019