Applications of Dynamical Systems in Biology and Medicine by Trachette Jackson & Ami Radunskaya

Author:Trachette Jackson & Ami Radunskaya
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

Our analysis of the phase model predicts that in this same network, two stable 8-cluster solutions exist with phase difference between nearest neighbors (see Table 2). In these solutions, each cluster contains 25 cells: C 0 = { 0, 8, 16, …, 192}, C 1 = { 1, 9, 17, …, 193}, C 2 = { 2, 10, 18, …, 194}, C 3 = { 3, 11, 19, …, 195}, …, C 7 = { 7, 15, 23, …, 199}. For the 8-cluster solution corresponding to , the cluster firing order is {C 0, C 3, C 6, C 1, C 4, C 7, C 2, C 5}. For , the clusters fire in the following order {C 0, C 5, C 2, C 7, C 4, C 1, C 6, C 3}. The phase difference between successively firing clusters is . See Figure 4(b).

There were some cases where the numerically observed stability of n-cluster solutions did not agree with that predicted by the phase model. While the 3-cluster solution was numerically stable in networks with N = 3p cells and single nearest neighbor coupling, with two nearest neighbor coupling, the maximal synaptic conductance g syn had to be weaker (0. 1 instead of 0. 2 mS/cm2) to numerically find these solutions. Additionally, recall that the phase model predicted that the 2-cluster solution is stable for all networks with an even number of neurons and either single or two nearest neighbor coupling. This solution corresponds to the network breaking into two clusters and , with phase difference ψ = π between nearest neighbors. We found this solution numerically in networks with single nearest neighbor coupling and N = 2, …, 200. However, with two nearest neighbor coupling we did not find this solution numerically. Closer consideration of the eigenvalues in this case shows that the stability is weaker, in the sense that the magnitude of the real part of the eigenvalues is smaller. Thus it is possible that the solution has a smaller basin of attraction and is harder to find numerically. Alternatively, the phase model may no longer accurately predict the network behavior as the coupling is too strong. In these networks, a different 2-cluster solution was found when N = 4p in which and . In this solution, the phase difference between every nearest neighbor is not the same. We discuss these 2-cluster solutions in more detail below.

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.