Figure 1

(Color online) Dynamics of coupled fast and slow positive feedback loops. (a) Schematic of the coupling in the models of Brandman et al. [2] and Zhang et al. [16]. The rate of synthesis of “output” is proportional to the sum of $A$ and $B$. “output” feeds back to enhance the synthesis of $A$ with a fast time constant ${\tau }_A$, and enhances the synthesis of $B$ with a slow time constant ${\tau }_B$. This schematic also describes multiplicative coupling [Eq. (7)]. (b) Response to a stimulus pulse for the model of Eqs. (1, 2, 3) (output synthesis depends on $A+B$) or the model of Eqs. (1, 2, 7) (output synthesis depends on $A\times B$). Stimulus amplitude (parameter $S$) is zero until $t=6.7\min$, at which time $S$ is set to either 1.5 or 2.5. At $t=15\min$ $S$ is reset to zero. Gaussian stimulus noise is always present with a standard deviation of 0.15 and an update time step of $1\mathrm{s}$. Model parameters have the standard values given after Eqs. (1, 2, 3) and Eq. (7).Reuse & Permissions

Figure 2

(Color online) Bifurcation in the model of Zhang et al. [16]. (a) Bifurcation diagrams for the bistable variants of the model of [Fig. 1a]. Steady states of $C_{\mathrm{out}}$ are traced as a function of a constant value of $S$. Equations (4, 5, 6) with their standard values were used to compute the $“A+B”$ curve, and Eqs. (4, 5, 7) were used for the $“A\times B”$ curve. “LP” denotes a limit point at which a steady state vanishes. Parameters have the standard values. (b) Bistability in the model of Zhang et al. [16] is preserved with internal stochastic noise. Equations (4, 5, 6) were used. The volume factor $\Omega =400$. Prior to choosing $\Omega$, all parameters were at standard values except $k_{\text{on}}=1.2\mu {\mathrm{M}}^{–1}{\mathrm{s}}^{–1}$, $k_{\min }=0.005\mu \mathrm{M}$, $k_{\min \mathrm{out}}=0.001\mu \mathrm{M}{\mathrm{s}}^{-1}$. Zero-order rate constants $(k_{\min },k_{\min \mathrm{out}})$ and the Hill constant $K$ are multiplied by $\Omega$. The upper bounds for $A$, $B$, and $C_{\mathrm{out}}$, which are 1.0 in Eqs. (4, 5, 6), are multiplied by $\Omega$. The second-order rate constant $k_{\text{on}}$ is divided by $\Omega$. Bistability for ${\tau }_A$ fast $\left(2\mathrm{s}\right)$ and ${\tau }_B$ slow $\left(200\mathrm{s}\right)$. The model is initialized in the lower state with $S=0.1$. At $t=33\min$, $S$ is increased to 0.5 for $7\min$. The model transits to the upper steady state. The $C_{\mathrm{out}}$ time course is over 20 simulations. For $B$, two time courses are shown. Time courses of $B$ are shown with ${\tau }_B$ slow $\left(200\mathrm{s}\right)$ and with ${\tau }_B$ fast $\left(2\mathrm{s}\right)$. (c) Simulations of the time course of fluctuation-induced escape from the lower state to the upper state. Each time course represents the evolution of the fraction $F_{\mathrm{trans}}$ of simulations that have transited at least once to the upper state, for an ensemble of 1000 simulations. Model parameters as in (b) except ${\tau }_B$ varies.Reuse & Permissions

Figure 3

(Color online) Dynamics of parallel uncoupled feedback loops. (a) Model schematic. The rate of synthesis of $C_{\mathrm{out}}$ is proportional to either the sum or product of $A$ and $B$. $A$ feeds back to enhance its own synthesis with a fast time constant ${\tau }_A$, and $B$ feeds back to enhance its own synthesis with a slow time constant ${\tau }_B$. (b) Simulated response of $A$ and $B$ to a stimulus. At $t=12\min$, $S$ is increased from 0 to 1.5, and remains elevated until $t=38\min$, at which time $S$ returns to 0. Gaussian stimulus noise is present with a standard deviation of 0.3 and an update time step of $1\mathrm{s}$. A increases to a plateau with large fluctuations, whereas $B$ increases slowly with small fluctuations. (c1) Simulated response of $C_{\mathrm{out}}$ to the stimulus of (b) with $A+B$. Rapid time course, $C_{\mathrm{out}}$ is driven mostly by changes in $A$ (${\lambda }_1=1.6$, ${\lambda }_2=0.4$). Slow time course, $C_{\mathrm{out}}$ is driven mostly by changes in $B$ (${\lambda }_1=0.4$, ${\lambda }_2=1.6$). (c2) Simulated response of $C_{\mathrm{out}}$ to the stimulus of (b) with $A\times B$. The system is no longer sensitive to differential changes to the coupling strength of the two loops.Reuse & Permissions

Figure 4

A bistable simplified model representing putative feedback loops in LTP and LTF induction and consolidation. (a) Model schematic. In a relatively fast feedback loop (time constant ${\tau }_A$), a kinase catalyzes (directly or indirectly) its own phosphorylation and activation. $A$ denotes the amount of active kinase. An external stimulus, such as an influx of ${\mathrm{Ca}}^{2+}$ into a neuron, would provide the initial kinase activation. In a second slow feedback loop (time constant ${\tau }_B$), A enhances the production of $B$ which in turn promotes the formation of $A$. $B$ could represent the total amount of kinase $(\text{inactive}+\text{active})$. (b) Bifurcation diagram illustrating the steady states of $A$ and $B$ as a function of a constant stimulus strength. Standard parameter values following Eqs. (11, 12) are used.Reuse & Permissions

Figure 5

(Color online) Resistance of elevated kinase activity to reversal by brief stimuli. (a) Development of resistance over time. At $t=2\mathrm{h}$, a $1\mathrm{s}$ elevation of $S$ to 200 induces a transition of $A$ to the upper state. $B$ increases over the next few hours. At $t=5\mathrm{h}$, a $100\mathrm{s}$ elevation of $k_{\mathrm{deg}A}$ from 1 to 11 fails to induce a state transition (upper $B$ and $A$ traces). $A$ drops briefly but returns immediately to the upper state. If the same brief elevation of $k_{\mathrm{deg}A}$ occurs $20\min$ after the transition to the upper state, then $A$ does transit back to the lower state, after which $B$ slowly declines (lower $B$ and $A$ traces). Model parameters are at standard values. (b) Bifurcation diagrams for $A$ as a function of $S$. The variable $B$ is treated as a fixed parameter, thus only Eq. (11) is used to compute the diagrams. Standard parameter values following Eqs. (11, 12) are used. Two bifurcation curves each show upper, middle, and lower steady states of $A$, for $B=3.26$ and $B=1.26$, respectively. (c) Stochastic simulation of resistance to depotentiation. $\Omega$ was chosen as 100. At $t=8\mathrm{h}$, a $1\mathrm{s}$ increase of $S$ to 200 drives an upwards state transition. The “average $A$” trace is over 20 simulations. Superimposed is a time course of $A$ for a single simulation. For all simulations, both the lower and upper state are stable. Parameter values differ somewhat from (a) because the lower steady state of (a) is not stable against internal noise for $\Omega \sim 100$. The changed parameter values are ${\tau }_B=3\mathrm{h}$, $K=0.3\mu \mathrm{M}$, $k_{\min A}=0.018\mu \mathrm{M}$, $B_{\mathrm{MAX}}=3.6\mu \mathrm{M}$, $k_{\min B}=1.2\mu \mathrm{M}$. At $t=17\mathrm{h}$, a $60\mathrm{s}$ increase of $k_{\mathrm{deg}A}$ from 1 to 3 fails to induce a state transition. However, the same brief increase in $k_{\mathrm{deg}A}$ applied shortly after the upwards transition returns $A$ to the lower state.Reuse & Permissions

Figure 6

Robustness of bistability against fluctuations as a function of the time constant ${\tau }_B$ and of the volume factor $\Omega$. Other parameters are as in Fig. 5c. Simulations are initialized in the lower steady state of $A$ and $B$. $S$ is constant at 0.1. Each time course represents the evolution of the fraction of simulations that have transited to the upper state for an ensemble of 1000 simulations. (a) Increase in stability with ${\tau }_B$. (b) Increase in stability with $\Omega$. Other parameters are as in (a), and ${\tau }_B$ is fixed at $100\mathrm{s}$. (c) Linear least squares fit to points denoting the natural logarithm of the transition rate constant as a function of $\Omega$.Reuse & Permissions