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Fig. 2 | The Journal of Mathematical Neuroscience

Fig. 2

From: Stable Control of Firing Rate Mean and Variance by Dual Homeostatic Mechanisms

Fig. 2

In neurons with fluctuating firing rate, dual homeostasis is effective under certain conditions. A firing rate unit receives a variable synaptic input \(I{(t)}\). The parameters are listed in Appendix 2. Vector fields of equation (5) in the \(({{x}}, g)\) phase plane are illustrated with arrows. The x- and g-nullclines are plotted with sample trajectories in the phase plane (above), and these sample trajectories are plotted over time (below). (A) If the target firing rate \(r_{{{x}} }\) of the intrinsic homeostatic mechanism is lower than the target firing rate \(r_{g }\) of the synaptic scaling mechanism (in this case, \(r_{{x}} = 2.5\) and \(r_{g} = 3.5\)), then the nullclines cross, and all trajectories are attracted to the fixed point at their intersection. (B) If \(r_{{{x}} } > r_{g }\) (in this case, \(r_{{x}} = 3.5\) and \(r_{g} = 2.5\)), then the nullclines do not cross, and g goes to zero. (C) If \(r_{{{x}} } > r_{g }\) and \(f_{{x}}\) is exchanged with \(f_{g}\) (\(r_{{x}} = 3.5\), \(r_{g} = 2.5\), \(f_{{x}}(r) = r^{2}\), \(f_{g}(r) = r\)), then the nullclines do cross, but the resulting fixed point is unstable

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