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Fig. 1 | The Journal of Mathematical Neuroscience

Fig. 1

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

Fig. 1

In neurons with constant firing rate, dual homeostasis fails to converge on a set point. A firing rate unit receives constant input \(I{(t)} = 1\). It is controlled by the homeostatic x (intrinsic homeostasis) and g (synaptic homeostasis) as described by (2) with \(f_{{x}}(r) = r\) and \(f_{g}(r) = r^{2}\). Other parameters are listed in Appendix 2. Vector fields of the control system are illustrated with arrows in the \(({{x}}, g)\) phase plane. 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 excitability-modifying 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 g increases and x decreases without bound. This phenomenon is called “controller wind-up.” (B) If \(r_{{{x}} } > r_{g }\) (in this case, \(r_{{x}} = 2.5\) and \(r_{g} = 3.5\)), then \(g\rightarrow0\), i.e., all afferent synapses are eliminated. (C) If \(r_{{{x}} } = r_{g }\) (in this case, \(r_{{{x}} } = r_{g }=3\)), then the nullclines lie on top of each other, creating a one-parameter set of fixed points that collectively attract all control system trajectories

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