Abstract
Molecular switches, such as the protein kinase CaMKII, play a fundamental role in cell signalling by decoding inputs into either high or low states of activity; because the high activation state can be turned on and persist after the input ceases, these switches have earned a reputation as ‘digital’. Although this binary perspective has been valuable for understanding synaptic plasticity over long timescales, accumulating experimental evidence suggests that molecular switches also control cellular processing on short timescales. To investigate this idea further, a non-autonomous, nonlinear ordinary differential equation, representative of a bistable molecular switch, is analyzed. The existence and uniqueness of model solutions to arbitrary input is proved for both the high and low states of activity. These results suggest that sub-state switch activity is an analog signal that tracks instantaneous input frequency, thereby increasing the capacity for information transfer to downstream cellular targets. Using simple dynamics based on the ubiquitous Hill equation, the model and theory make intriguing predictions about synaptic plasticity and suggest a multiplexed encoding of instantaneous frequency information over short timescales, with integration of total activity over long timescales, helping to reconcile contrasting perspectives presented in the literature.
Introduction
Many different cellular inputs lead to transient changes in cytosolic calcium (Ca2+) levels, generating temporally complex signals that reflect a wealth of information [1]. As such, cells express highly conserved molecular decoders capable of translating Ca2+ oscillations into downstream signalling events that affect diverse processes such as gene transcription, development and aging, neural network homeostasis and the synaptic plasticity that underlies learning and memory [2–9]. A celebrated example of a Ca2+ decoder is the bistable molecular switch Ca2+/calmodulin (CaM)-dependent protein kinase II (CaMKII; Box 1), which is driven by transient levels of cytosolic Ca2+ into either high or low states of activity. When stabilized through negative regulation by phosphatases, self-exciting kinases such as CaMKII are an ideal component of signal amplification and have been previously likened to transistors on a computer chip, in that they may be turned on or off, presenting an ideal substrate for computation in cellular systems [10]
The classic CaMKII experiments of De Koninck and Schulman provided the first demonstration that a molecular switch can decode the frequency of periodic Ca2+ pulses into distinct levels of persistent activation [11]. Subsequent modelling of CaMKII autophosphorylation dynamics captured this hysteresis effect, that is, the ability of the high activation state to persist beyond the original Ca2+ signal and act enzymatically over long timescales [12–14]. In these studies, the relationship between Ca2+ concentration and the state of the molecular switch was determined from simulations of large, parameterized systems of differential equations that are not readily amenable to deeper mathematical analysis; furthermore, these studies were restricted to periodic inputs. One notable exception is the work of Graupner and Brunel, who developed a reduced Ca2+ based model of long timescale postsynaptic plasticity [15]. In order to better understand frequency coding over short timescales, this article analyzes a reduced description of molecular switch behaviour when subject to general aperiodic forcing and in the presence of noise. As the study of cellular information processing shifts from individual transduction pathways, toward the emergent properties of complex signalling networks, simple mathematical models are becoming indispensable tools for both experimentalist and theoreticians alike [16, 17].
Box 1: The bistable molecular switch CaMKII and synaptic plasticity
Accounting for approximately 1-2% of all brain protein, the molecular switch CaMKII is a central hub of cell signalling networks and can exert both pre‐ and post-synaptic control over information transmission in the central nervous system [18]. Once bound to the Ca2+-CaM complex, the kinase’s ability to cooperatively autophosphorylate produces two distinct stable states: either high or low levels of enzymatic activation.
Postsynaptically, after repetitive stimulation, the high activation state may persist long after the calcium signal subsides and can strengthen the connection between neurons, for example, the hippocampal CA3-CA1 synapses that support learning and memory [19]. Presynaptically, CaMKII also modifies connection strength [20–22]. At mouse hippocampal CA3-CA1 synapses, knocking-out the αCaMKII isoform leads to reduced synaptic potentiation under paired pulse facilitation protocols when compared to the wild-type [23]. Through enzymatic phosphorylation of voltage gated Ca2+ channels and ryanodine receptors, αCaMKII can enhance Ca2+ entry and Ca2+-induced Ca2+ release in response to high frequency signals [24]. However, at the same CA3-CA1 synapses, post-tetanic potentiation protocols generate enhanced levels of potentiation in the knock-out mice, illustrating that αCaMKII may also limit neurotransmitter release depending on the frequency and duration of the input [23]. Furthermore, a frequency dependent shift from paired pulse facilitation to paired pulse depression has been reported [25] and αCaMKII has been shown to serve as a negative, activity-dependent regulator of neurotransmitter release probability [26]. This effect can be partially explained by the fact that CaMKII phosphorylates Ca2+-activated potassium channels that hyperpolarize the presynaptic terminal [27], decreasing the likelihood of Ca2+ entry and neurotransmitter release. Intriguingly, αCaMKII also plays a non-enzymatic role in presynaptic CA3-CA1 plasticity by regulating the number of docked synaptic vesicles containing neurotransmitter [28]. In this case, decreased transmitter release is likely explained by the fact that αCaMKII is acting as a sink for intracellular Ca2+, lowering the cytosolic levels that drive the machinery of synaptic vesicle fusion and influencing the size of the readily releasable vesicle pool [29, 30].
One of the most influential discoveries about CaMKII is its ability to decode the frequency of periodic Ca2+ pulses into distinct amounts of long lasting, autonomously activated kinase [11]. However, the interpretation of CaMKII as a frequency decoder has been criticized based on the fact that mean values of activity, evoked by different combinations of Ca2+ pulse size, duration and frequency, are ambiguously mapped into the same level of autonomously activated switch [31]. To address this criticism and bridge our understanding of CaMKII function over short and long timescales, this article investigates whether the concentration of activated switch acts as a reliable, sub-state analog signal that encodes frequency information over short timescales, where Ca2+ pulse size and duration are stable [32]. The experimental evidence discussed above suggests that frequency coding by these ‘digital’ molecular switches is more sophisticated than previously thought and that fluctuations in presynaptic αCaMKII activity meaningfully decode instantaneous frequency information, translating it into bidirectional, real-time control of synaptic strength.
Results
A Bistable Switch Model
The following differential equation is an abstraction of a bistable molecular switch and was originally proposed as a model of genetic development by Lewis et al. [33]. This relatively simple model is a useful analytical tool to understand the general properties of bistable kinetic systems; although the model interpretation and results are centered on CaMKII and synaptic plasticity, the reader is encouraged to consider the broader implications for instantaneous frequency coding with molecular switches (e.g., mitogen-activated protein kinases).
In this formulation, the level of activated CaMKII (y) is stimulated by the presence of Ca2+ bound to CaM (s), which will be studied as a function of time. For simplicity we assume that pulses of Ca2+ are bound upon cell entry, which is reasonable since CaM is found in large concentrations surrounding Ca2+ channels and has a strong affinity for Ca2+ [34]. Deactivation is directly proportional to the active CaMKII concentration at a rate k1, representing the collective activity of protein phosphatases. Finally, once activated, CaMKII has the ability to cooperatively bind Ca2+-CaM and autophosphorylate itself, which motivates the nonlinear, positive feedback term captured by the Hill equation, where k2 and k3 are the association and dissociation constants respectively. Due to physiological constrains, y, s, k0, k1, k2, k3 ≥ 0.
This generic model has been previously applied to genetic networks [33, 35, 36], transcriptional regulation [37, 38], mitogen-activated protein kinases [39], and incorporated into a larger phenomenological model of presynaptic plasticity [40]. Although insightful for their specific systems, these studies retain a large numbers of parameters that clutter analysis and obscure the generality of the results. Therefore, it is desirable to reduce the number of parameters and facilitate the following analysis by performing routine nondimensionalization. Let and which, when substituted into the original equation and simplifying gives the reduced but dynamically equivalent form:
This article is interested in a time varying c c0 + cl (ρ), where c0 reflects residual cytosolic Ca2+ whose slow dynamics are treated as fixed on the fast timescales over which the local Ca2+ signal cl (ρ) fluctuates [41]. The scale factor , the quotient of the switch deactivation and activation parameters, will be reintroduced later in order to connect the switch dynamics to time in seconds and stimulation frequency in Hz. Finally, for highly cooperative reactions, n = 2 is a reasonable approximation [42] and a convention maintained by all of the studies listed above. The following bifurcation analysis is illustrated for n = 2, which allows for exact analytical solutions (Fig. 1 and Methods); however, the main results are then generalized to arbitrary n ∈ ℝ+, which is much more realistic and has important consequences for frequency coding.
Stability and Bifurcation Analysis
Although interested in frequency-driven fluctuations, we must first examine the fixed state behaviours of the bistable switch. An important reason for reducing the number of model parameters is to simplify the analysis of how the behaviour of the system changes as a function of the parameter values. For Equation 1, having selected n = 2, we only need to consider the effect of varying r and c; depending on their values, we may have one, two or three equilibrium points (x*), where the rate of change of the switch is equal to zero. For example, consider the values r = 0.52 and c = 0.04 that support bistability: there are three fixed points, two of which are stable, as illustrated by the switch’s potential function (Fig. 1A). As r and c change, saddle node bifurcations can occur, resulting in the presence of only the high or low activation state. The corresponding bifurcation diagrams are displayed in Figure 1B; their derivation is found in the Methods section.
A key feature of bistability is the hysteresis effect, where the same value of a parameter may evoke different states depending on the history of activity. For example, as c increases, x* grows larger until crossing the rightmost cc, where a saddle node bifurcation occurs and the switch jumps up to the high activation state as the low state disappears (Fig. 1B, i). Now, as c decreases back into the bistable range, the high activation state is preserved, and only lost when c crosses below the leftmost value of cc. This history dependent behaviour is presumably central to CaMKII activity and the synaptic plasticity that underlies learning and memory (Box 1) [19]. A similar phenomenon occurs for the negative regulation parameter r (Fig. 1B, ii). The values of rc and cc are plotted parametrically as a function of the active switch in the bifurcation curves (Fig 1B, iii). The bifurcation surface summarizes this information completely (Fig. 1B, iv).
Existence of Sub-state Solutions
To date, studies of Lewis et al.’s model have been restricted to static input and periodic forcing. It is of principal interest to characterize the model behaviour in response to aperiodic forcing, in order to gain a more general, physiologically realistic understanding of frequency coding with molecular switches. In addition to coding frequency information into stable levels of activated switch for minutes to hours [11, 19], what about frequency coding on the order of milliseconds to seconds? In a region surrounding a stable activation state, is there a unique sub-state solution for a time varying input signal? This question is not trivial, since small changes in the initial conditions of a nonlinear system (i.e., past switch activity) may generate drastically different behaviours. Understanding the relationship that determines whether solutions converge or diverge provides valuable insight into the properties of bistable molecular switches.
We now reintroduce the scale factor T since, in the following section, we are interested in studying frequency in Hz and time in seconds (t). As such, Equation 1 becomes
First, to establish the existence of solutions around the high and low switch states, consider Equation 2 and note that f explicitly depends on the time-varying forcing term, c(t) c0 + cl (t). The function f (t, x(t)) is assumed to be Lipschitz continuous and well-defined within strips, y− ≤ x(t) ≤ y+ satisfying the conditions f (t, y−) > 0 and f (t, y+) < 0 ∀ t ℝ+, which trap solutions within these boundaries. For (c, r) corresponding to the bistable region of parameter space (Fig. 1B, iii), there exist two infinite strips, x(t) ∈ (yl−, yl+) and x(t) ∈ (yh−, yh+), each surrounding one of the stable equilibrium points (x*). Now, we wish to locate values for the low state (yl−, yl+) and high state (yh−, yh+), where the existence of local time-varying solutions can be guaranteed. This problem is intimately linked to bifurcation, since yl+ and yh− depend on the values of c and r. The choice of a lower bound for the strip that exists around the low activation state is yl− = 0, since the physiological restriction c(t) ≥ 0 implies f (t, 0) > 0 ∈t, ignoring the degenerate case of c(t) = x(t) = 0. The upper bound of the lower strip, yl+, should occur at a value , left of the unstable equilibrium where , such that this condition ensures that the system is not trivially displaced into the up-state by a single Ca2+ pulse with amplitude Δc. For the high concentration strip (yh−, yh+), the lower bound yh− is chosen as a value of x infinitesimally greater than , that is, . Since we have restricted r and c > cc (leftmost) to the bistable range, we know that f (t, yh−) > 0 ∀t. For the upper bound of the high activation strip, it is enough to note that for and, since we wish to maximize the width of the strip, we take x arbitrarily large, denoting this value by yh+ = x∞. During stimulation, if (c, r) drifts out of the bistable region of parameter space, a saddle node bifurcation occurs and only one infinite strip exists; in this case, y− = 0 and y+ = x∞.
By invoking the Cauchy-Peano theorem, we guarantee the existence of at least one sub-state solution for every initial condition found within the strip regions defined above, since the conditions on the sign of the derivative f (t, x(t)) define trapping regions. However, this theorem says nothing about whether solutions starting at different initial conditions will converge to a unique, stimulus-driven response and track the instantaneous frequencies of the input signal.
Uniqueness of Sub-state Solutions
As motivation for the following results, Figure 2A shows an example switch response to an 8 Hz Poisson pulse sequence, which is convolved with an alpha function filter (30 ms; Methods), then normalized to the signal’s maximum and scaled by Δc = 0.5 to create a representative input signal, which the switch tracks closely.
We now establish the stability and uniqueness of solutions in each strip for distinct initial conditions. Consider a general infinite strip (y−, y+), where x(t) is a solution to Equation 2 with initial condition x0 ∈ (y−, y+). Assume there is another solution, u(t), with a different initial condition u0 ∈ (y−, y+). Writing z(t) = | u(t) − x(t) | and first assuming n is a positive integer, we see that
The expression p(u, x, n) reflects a tendency for solutions to diverge and achieves a maximum at an intermediate switch level that separates the low and high states of activation. Now, consider p(u, x, n) for the special case of n = 2, used in the bifurcation analysis; in this case, which is plotted in Figure 2B. Setting the partial derivatives of the function to zero and solving for u and x, yields a critical point: . Substituting this into p gives a global maximum of . Since , we can apply Grönwall’s inequality, which gives us the following:
Substituting the expression for z(t) and solving this integral exponent yields, and, as t → ∞, we have
For , we obtain
By the squeeze theorem we conclude that | u(t) − x(t) |→ 0 as t → ∞. Therefore, a unique frequency-driven solution exists within either strip and is independent of the initial conditions. The time taken to converge to the unique solution is inversely proportional to T (Fig. 2C). The value T = 0.01 seconds was chosen here for the specific example switch, CaMKII, whose dissociation constant (k3) is at least 100-fold smaller than the activation constant (k2) that governs the rate of autophosphorylation [43]. Unlike the larger value of T 0.1 seconds, T = 0.01 permits quick convergence and reliable encoding for the action potential frequencies characteristic of hippocampal CA3-CA1 synaptic input (approximately 1-15 Hz) [44]. Smaller values of T permit rapid convergence and more sensitive frequency coding, but may become overly sensitive to temporary lulls in activity when c briefly drops below the leftmost cc (recall Fig. 1B, i).
It should be noted that r > 0.65 is an absolute guarantee of convergence to a unique frequency driven solution; but, from the bifurcation analysis (Fig. 1B, iii; Methods), we know that bistability does not exist for this value. However, in general, only −r + p(u, x, 2) < 0 is required, which, for low and high concentrations of activated switch, is obtained at smaller values of r that support bistability. Although a unique encoding of sub-state solutions still exists for smaller r values, convergence about the low activation state is vulnerable to perturbation by short Ca2+ inter-pulse intervals, thus acting as a high frequency event (burst) detector through induction of long term switch activation (i.e., hysteresis; Fig. 2D). In theory, this dynamic threshold (the separatrix), is sensitive to recent levels of activation, and could be purposefully modulated by the cell through regulation of the parameters r and c0, that is, the expression of protein phosphatases and residual levels of cytosolic Ca2+ [45]. To restore the low state, the cell simply needs to adjust c0 to fall below the leftmost critical value cc. Note, in this simulation, the kernel was specifically chosen to be 30 ms based on literature values for the time course of local synaptic Ca2+ signals [15, 46, 47].
Realistically, the Hill function exponent need not be restricted to integer values, which is unlikely in real biological systems. Thus, in the above proof, the expression p(u, x, n) is left as for n ∈ℝ+, since there is no longer a closed form expression for the factorization of the numerator by u − x. The function p(u, x, n) has only one critical point at u = x, which occurs at an apparent discontinuity due to the factor u − x in the denominator. However, assessing the limit as the difference between x and u becomes infinitesimally small, making the change of variable u = x + h as h → 0, and recognizing the limit definition of the power rule for differentiation, yields an expression for the maximum of p(u, x, n) ∀u, x, n ∈ ℝ+:
For each value of the exponent n, the global maximum of this expression is determined for ∀x ∈ℝ+, and plotted (Fig. 2E). For n > 0.012, the minimum of the class of functions p(u, x, n) is found at n = 1.55. Fascinatingly, the αCaMKII isoform was reported by De Koninck and Schulman to have a Hill function exponent of 1.6 [11]; although their Hill function argument was Ca2+-CaM, this striking match suggests that αCaMKII’s activation function may operate with this particular exponent as it provides the minimum level of negative regulation required to maintain convergence of unique input driven switch activity, even at intermediate levels of the switch response, where r must be stronger to guarantee unique solutions about the low activation state. This has the putative benefit of minimizing the amount of broadly acting phosphatase that may interfere with a cell’s other phosphorylation-dependent processing. The reader should note that this result is independent of the model parameters, suggesting it is a very general property of the Hill function and regulated, self-excitatory biological phenomena.
Molecular Switches and Stochastic Resonance
If the model is to capture actual molecular switch behaviour in vivo, then we must understand frequency coding in the presence of noise. The results presented in this section are generated by Equation 2 with additive Ornstein-Uhlenbeck noise, η(t), which evolves according to the stochastic differential equation where ξ(t) is bounded Gaussian noise, N (0,1), whose amplitude is scaled by a parameter σ choice of the time constant ρn is based on previous studies of noisy microdomain Ca2+ fluctuations, where an upper bound for the autocorrelation time was determined to be approximately 10 ms [48, 49]. This choice has the added benefit of matching our switch time constant T, should we instead assume the noise is inherent to switch activation. Figure 3A shows the power spectrum (Pc) of a weak sinusoidal calcium oscillation, c = c0 + αsin(2πφt), where c0 = 0.04, α =0.02 and φ = 2 Hz, which was selected based on the mean action potential frequency associated with the CA3 and CA1 regions of the hippocampus [50]. As expected, the noisy switch oscillates at the frequency φ, reflected in its power spectrum (Px). Given our interest in frequency transfer, it is natural to ask whether noise improves or degrades the switch’s frequency coding ability. Very recently, the model of Lewis et al., studied under the context of genetic regulation with n = 2, has been shown to produce the stochastic resonance effect [38], which is confirmed here (Fig. 3B). As σ increases from 0, frequency transfer, measured as the ratio of switch power to signal power at φ, dips slightly and then improves dramatically, achieving a maximum at 0.29, followed by a quick decrease as the noise becomes dominant. When changing the exponent from n = 2 to n = 1.6, this spectral amplification becomes significantly larger, suggesting again that presynaptic αCaMKII functions as a powerful frequency decoder and that the exponent n = 1.6 has evolved to optimize this purpose. The reader should note that, for comparisons sake, r = 0.65 and r = 0.61 were selected respectively for n = 2 and n = 1.6 based on Fig. 2E, but that this effect is robust to changes in r and <. Setting n = 1.6 also shifts the optimal noise strength to a substantially lower value, 0.09, which has the putative benefit of harnessing stochastic resonance at lower levels of noise that could otherwise be detrimental to different cellular operations.
The results of Kang et al. [38] depend on a full complement of parameters, which begs the question of whether stochastic resonance is a generic feature of the model switch or whether the effect is only significant for a certain range of the parameters. The dimensional reduction of the switch model performed here allows this question to be easily addressed as a function of the parameters c0, r and n. Figure 3C shows that the parameter r has significant influence over the value of σ that produces optimal spectral amplification and that, for some combinations of c0, r and n, the stochastic resonance effect disappears completely. The absence or presence of stochastic resonance may prove useful for deducing parameter ranges of molecular switches in vitro and in vivo. Furthermore, these noise fluctuations drive state transitions and, depending on their intensity (σ), may generate unimodal or bimodal distributions of switch activation (Fig. 3D); this provides another experimentally testable prediction for αCamKII, given that the switch state controls neurotransmitter release probability (Box 1).
Long timescale switch activation
A potential caveat of the bistable switch model is that, even in the autonomous high activation state, the population of phosphorylated units (x) are still subject to the phosphatase activity (r). This places difficult constraints on the cell for long-timescale activation: if c0 and r are not controlled carefully, the upstate can be lost. The model effectively represents all of the phosphorylated subunits in a population of CaMKII molecules (each having twelve phosphorylation sites). When one of these dodecamers becomes fully phosphorylated, it effectively becomes impervious to negative regulation by the phosphatases, since any cleaved subunit will immediately be re-phosphorylated by its neighbouring subunits and the enzyme can be shielded by its interactions with downstream targets [19]. Until now, the work presented here has ignored this feature of CaMKII dynamics. Therefore, we introduce a new variable to represent the level of autonomously activated switch that persists after the stimulus has been removed, even when Ca2+ levels drop below the leftmost critical value cc that supports hysteresis (Fig.1B, i). Inspired by the work of Pinto et al. [31] (Box 1), we assume that the total amount of autonomously activated switch (X) is simply proportional to the average amount of input and thus, the amount of phosphorylated switch x(t) that occurs over the duration of stimulation, Δt :
For simplicity, a basal rate of transition to the fully autonomous state, ω, is assumed. Figure 3E shows the amount of autonomously activated switch in response to repeated realizations of Poisson input over a range of frequencies. Note that the amount of autonomously activated switch performs a logarithmic compression of the average frequency over the course of stimulation, while the instantaneous levels of phosphorylated switch track the instantaneous input frequencies throughout (as in Fig. 2A). As a final validation of the model’s ability to produce CaMKII-like behaviour, the essence of De Koninck and Schulman’s experimental results ([11]; see Fig. 4 within) and the model of Dupont et al. ([12]; Fig. 3E within) are captured qualitatively by Equations 2 and 3 (Fig. 3F). This required setting T on the order of 10−1, which may reflect altered kinetics under the artificial conditions of their experiment, or the need for further refinement of the model presented here. For instance, the proportion ω is expected to grow larger as more of the subunit population becomes phosphorylated and cooperative activation grows stronger [51, 52], leading to an increased likelihood for individual dodecamers to transition to the fully autonomous state. This is expected to improve the reproduction of De Koninck and Schulman’s results by flattening the curves at lower frequencies and steepening them at higher frequencies [11]. Future work should seek to determine ω(x), with the hopes of identifying reduced representations of strongly nonlinear CaMKII activation.
Discussion
A main goal of this study was to extend the frequency coding idea of De Koninck and Schulman [11] in a generic switch model that captures the qualitative behaviour of CaMKII. The model of Lewis et al. [33], helps to reconcile contradictory perspectives of CaMKII function [11, 31] and suggests dual streams of information transfer that are temporally multiplexed: over short timescales, where the size and duration of the Ca2+ pulse is expected to be stable [32], the molecular switch acts as an encoder of instantaneous frequency information and apparently functions to bidirectionally regulate transmitter release probability at synapses through a combination of enzymatic and non-enzymatic activity (Box 1). Over longer timescales, the switch integrates signal intensity, which dictates long term changes in synaptic strength and is dependent on multiple factors such as slow Ca2+-induced Ca2+ release (affecting c0) [24, 53], the size of the Ca2+ pulse, its duration and the mean frequency of stimulation. These latter features are expected to adjust the frequency sensitivity of the system by differentially modulating the relative activation of protein phosphatases [45], which influence the threshold for detecting high frequency events, such as bursts of action potentials [54], that could drive up the switch activation state and influence release probability after the input subsides. Furthermore, phosphatases (r) have been treated statically in this study but actually have a high affinity for Ca2+ and their dynamics will additionally contribute to frequency tuning and neurotransmitter release [55–57]. Importantly, the work presented here provides some testable predictions for synaptic physiologists: establishing the presence of both bimodal and unimodal synaptic release that depends on αCaMKII and noise, as well as studying the real-time modulation of release probability at central synapses by αCaMKII in response to natural, aperiodic stimulation patterns (particularly bursting).
Bistable molecular switches are a conserved feature of cell signalling networks and generate combinatorial power in their collective action [58–60]. Much in the way that the leaky-integrate and fire model has been a successful abstraction of neuronal activity, providing a trade-off between performance and a reduced description that facilitates network studies [61, 62], the model of Lewis et al. captures the core essence of molecular switches while remaining amenable to mathematical analysis. The relative simplicity of the model and its application to diverse signalling pathways make it a useful framework for further theoretical and experimental investigations into signalling networks and cellular computation.
Methods
Bifurcation Analysis
The first step of the bifurcation analysis is to find the equilibrium points. Setting n = 2, we rewrite Equation 1 as, where and h(x) = rx − c. The fixed points occur when g(x) − h(x) = 0, which amounts to finding the solutions of the polynomial −rx3 + (c +1)x2 − rx + c = 0. First, fix c and examine the effects of varying r. When c = 0, x = 0 is a fixed point, and, for a particular range of r, there exists two other positive valued fixed points, given by the roots of −rx2 + x − r = 0. The critical value of the parameter r, denoted by rc is found by setting g(x) = h(x) and g′(x) = h′(x), which, when solved, gives . Three values of x satisfy this equality: −1, 0 and 1. Since we are not considering negative values of x, we have two critical points, r = 0 and . Therefore, when c = 0, the system is bistable for . For c > 0, r can be larger than while still preserving bistability (as in Fig. 1A). We know rc occurs when h(x) = g(x) and h′(x) = g′(x); therefore, when h(x) > g(x) we lose a fixed point through a saddle node bifurcation. For x > 0, the maximum of g(x) is found at which gives . Therefore, when , only one fixed point exists
Now, we are interested in fixing r and examining the effects of varying c. To find cc we set g(x) = h(x) and g′(x) = h′(x), which gives and . Substituting the first expression into the second, we get . We differentiate with respect to x in order to locate the maximum value for . This gives x = 0 and , which corresponds to cc = 0 and . When c > c, only one fixed point exists for all values of r. For a fixed value of r that supports bistability, as c increases from 0 and crosses a critical value (cc), the fixed point x* will jump up to the high amplitude branch. If c is now decreased, the fixed point remains on the high amplitude branch even as c becomes smaller than the corresponding cc. This hysteresis effect permits switch activation to remain as the transient Ca2+ signal subsides, consistent with the findings from synaptic plasticity experiments (Box 1). Using the expressions derived for the critical values of rcand cc, we plot them parametrically as functions of x (Fig. 1B, iii). Saddle node bifurcations occur all along the boundary of these curves, it is here we find the values of r and c for which only two fixed points occur. Crossing each branch results in a pairwise collision and disappearance of two fixed points. Note where the bifurcation curves meet tangentially, , here we observe a codimension two bifurcation; beyond this point there is only one fixed point and the distinction between low and high activation states is blurred (Fig. 1B, iii).
Computational Specifications and Miscellaneous Details
Simulations were solved using the 4th order Runge-Kutta method, with the exception of the Ornstein-Uhlenbeck noise, which was solved using the stochastic Euler method (time step of 1 ms in all cases). All simulations were performed using custom code, available upon request to the author, and were implemented on a Linux machine running Ubuntu 16.04 with an Intel core i7-6700 CPU, 3.4 GHz processing speed, and 62 GB of RAM.
Pulse train sequences {ti} were convolved with the filter , whose decay constant σc was set to 30 ms, reflecting an accommodation of both pre‐ and post-synaptic calcium decay values from the literature that range from 15-43 ms [15, 46, 47]. The resulting input signal was normalized to the maximum value and then scaled by Δc. The decay value is closely related to the input frequencies typical of a given synapse and the definition of what constitutes a high frequency event in the system, since for events occurring faster than the decay, Ca2+ accumulates quickly, driving the switch into the upstate. The putative burst detector will work for different σc, but may require a different set of corresponding switch parameters, range of stimulation frequencies and pulse amplitudes.
Histogram bin sizes for Fig. 3D were set using the Freedman-Diaconis method [63].
Acknowledgements
I would like to thank Lin Wang for useful discussions and input during the early stages of the project, as well as Leonard Maler and Richard Naud for proofreading the manuscript and providing valuable comments. This work was funded by training grants awarded to S.E.C. from both the National Sciences and Engineering Research Council of Canada and the Canadian Institute of Health Research