Fig. 4

Convergence of intrinsic/synaptic dual homeostasis is compromised by short homeostatic time constants and temporally correlated noise. Firing rate r is described by equation (15) with parameter values listed in Appendix 3. (A)-(B) The system simulated for Fig. 1A-D is modified by reducing homeostatic time constants by a factor of 50: we set \(\tau_{{{x}}} = 10~\mbox{s}\) and \(\tau_{g} = 1\text{,}000~\mbox{s}\). Trajectories enter and remain within a large neighborhood of the fixed point observed in Fig. 1A, but fluctuate randomly within that neighborhood. By Lemma 1 these trajectories converge in the small-ϵ limit, so this neighborhood represents the ball of radius \(\alpha(\epsilon)\) that traps all trajectories and shrinks to zero as \(\epsilon\rightarrow0\). (C)-(D) The system described in Fig. 1A is modified by introducing long temporal correlations into the time course of the input current: \(I{(t)}\) is an Ornstein-Uhlenbeck process described by the SDE \(\tau_{I} \, dI = -I\, dt + d\xi\), where ξ is white noise with unit variance and \(\tau_{I} = 10~\mbox{s}\). Again, trajectories enter and remain within a large neighborhood of the fixed point observed in Fig. 1A but fluctuate randomly within that neighborhood