Abstract
We analytically investigate the stability of splay states in the networks of N globally pulse-coupled phase-like models of neurons. We develop a perturbative technique which allows determining the Floquet exponents for a generic velocity field and implement the method for a given pulse shape. We find that in the case of discontinuous velocity fields, the Floquet spectrum scales as and the stability is determined by the sign of the jump at the discontinuity. Altogether, the form of the spectrum depends on the pulse shape, but it is independent of the velocity field.PACS: 05.45.Xt, 84.35.+i, 87.19.lj.
Highlights
The first objective of network theory is the identification of asymptotic regimes
We develop a perturbative analysis for the stability of splay states in ensembles of N globally pulse-coupled identical neurons
The technical details of some lengthy calculations have been confined in the appendices: Appendix A is devoted to the derivation of the splay state solution; Appendix B contains the derivation of the leading term of the period T for the LIF model; Appendix C is concerned with the linear stability analysis
Summary
Substituting the μk expansion (40) and retaining the leading terms, we obtain Eq (41). By inserting the resulting expansion for U (1/N ) into the expressions for F 1 and F 1, we obtain, respectively,. ), neglecting higher orders because they contribute to the definition of variable and we need terms at least of order The previous expansions in Eq (41), we obtain a closed equation for the eigenvalues and eigenvectors, eiφk (0) + (1) − (0) + (0) (1) 1 N. where we have introduced the shorthand notation B in order to characterise a term of order. ), whose explicit expression is not necessary, since it turns out to contribute to the definition of the variable, and it is, one order beyond what we need.
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