Toward a microscopic model of bidirectional synaptic plasticity

Random Walk Approximation for Stochastic Processes on Graphs

We introduce the Random Walk Approximation (RWA), a new method to approximate the stationary solution of master equations describing stochastic processes taking place on graphs. Our approximation can be used for all processes governed by non-linear master equations without long-range interactions and with a conserved number of entities, which are typical in biological systems, such as gene regulatory or chemical reaction networks, where no exact solution exists. For linear systems, the RWA becomes the exact result obtained from the maximum entropy principle. The RWA allows having a simple analytical, even though approximated, form of the solution, which is global and easier to deal with than the standard System Size Expansion (SSE). Here, we give some theoretically sufficient conditions for the validity of the RWA and estimate the order of error calculated by the approximation with respect to the number of particles. We compare RWA with SSE for two examples, a toy model and the more realistic dual phosphorylation cycle, governed by the same underlying process. Both approximations are compared with the exact integration of the master equation, showing for the RWA good performances of the same order or better than the SSE, even in regions where sufficient conditions are not met.

Energy Consumption and Entropy Production in a Stochastic Formulation of BCM Learning

In a series of previous studies, we provided a stochastic description of a theory of synaptic plasticity. This theory, called BCM from the names of the three authors, has been formulated in two ways: the original formulation, where the plasticity threshold is defined as the square of the time-averaged neuronal activity, and a newer formulation, where the plasticity threshold is defined as the time average of the square of the neuronal activity. The newest formulation of the BCM rule of synaptic activity has interesting statistical properties, derived from a risk (or energy) function, the minimization of which leads to seeking of interesting projections in high-dimensional space. Moreover, these two rules, if implemented by a chemical master equation approach, show another interesting difference: the original rule satisfies the detailed balance, whereas the other not. Based on this different behavior, we found a continuous parameterization between these two rules. This parameterization shows a minimum that corresponds to maximum negative eigenvalues of the Jacobian matrix. In addition, the newest rule, due to the fact that it is in a nonequilibrium steady state (NESS), shows a higher level of plasticity than the original rule. This higher level of plasticity has to be interpreted in the framework of open thermodynamical systems and we show that entropy production and energy consumption in the newest rule are both less than in the original BCM rule.

Simplified calcium signaling cascade for synaptic plasticity

We propose a model for synaptic plasticity based on a calcium signaling cascade. The model simplifies the full signaling pathways from a calcium influx to the phosphorylation (potentiation) and dephosphorylation (depression) of glutamate receptors that are gated by fictive C1 and C2 catalysts, respectively. This model is based on tangible chemical reactions, including fictive catalysts, for long-term plasticity rather than the conceptual theories commonplace in various models, such as preset thresholds of calcium concentration. Our simplified model successfully reproduced the experimental synaptic plasticity induced by different protocols such as (i) a synchronous pairing protocol and (ii) correlated presynaptic and postsynaptic action potentials (APs). Further, the ocular dominance plasticity (or the experimental verification of the celebrated Bienenstock-Cooper-Munro theory) was reproduced by two model synapses that compete by means of back-propagating APs (bAPs). The key to this competition is synapse-specific bAPs with reference to bAP-boosting on the physiological grounds.

Insulin Signaling in Insulin Resistance States and Cancer: A Modeling Analysis

Insulin resistance is the common denominator of several diseases including type 2 diabetes and cancer, and investigating the mechanisms responsible for insulin signaling impairment is of primary importance. A mathematical model of the insulin signaling network (ISN) is proposed and used to investigate the dose-response curves of components of this network. Experimental data of C2C12 myoblasts with phosphatase and tensin homologue (PTEN) suppressed and data of L6 myotubes with induced insulin resistance have been analyzed by the model. We focused particularly on single and double Akt phosphorylation and pointed out insulin signaling changes related to insulin resistance. Moreover, a new characterization of the upstream signaling of the mammalian target of rapamycin complex 2 (mTORC2) is presented. As it is widely recognized that ISN proteins have a crucial role also in cell proliferation and death, the ISN model was linked to a cell population model and applied to data of a cell line of acute myeloid leukemia treated with a mammalian target of rapamycin inhibitor with antitumor activity. The analysis revealed simple relationships among the concentrations of ISN proteins and the parameters of the cell population model that characterize cell cycle progression and cell death.

Memory maintenance in synapses with calcium-based plasticity in the presence of background activity

Most models of learning and memory assume that memories are maintained in neuronal circuits by persistent synaptic modifications induced by specific patterns of pre- and postsynaptic activity. For this scenario to be viable, synaptic modifications must survive the ubiquitous ongoing activity present in neural circuits in vivo. In this paper, we investigate the time scales of memory maintenance in a calcium-based synaptic plasticity model that has been shown recently to be able to fit different experimental data-sets from hippocampal and neocortical preparations. We find that in the presence of background activity on the order of 1 Hz parameters that fit pyramidal layer 5 neocortical data lead to a very fast decay of synaptic efficacy, with time scales of minutes. We then identify two ways in which this memory time scale can be extended: (i) the extracellular calcium concentration in the experiments used to fit the model are larger than estimated concentrations in vivo. Lowering extracellular calcium concentration to in vivo levels leads to an increase in memory time scales of several orders of magnitude; (ii) adding a bistability mechanism so that each synapse has two stable states at sufficiently low background activity leads to a further boost in memory time scale, since memory decay is no longer described by an exponential decay from an initial state, but by an escape from a potential well. We argue that both features are expected to be present in synapses in vivo. These results are obtained first in a single synapse connecting two independent Poisson neurons, and then in simulations of a large network of excitatory and inhibitory integrate-and-fire neurons. Our results emphasise the need for studying plasticity at physiological extracellular calcium concentration, and highlight the role of synaptic bi- or multistability in the stability of learned synaptic structures.

Synaptic plasticity is widely believed to be the main mechanism underlying learning and memory. In recent years, several mathematical plasticity rules have been shown to fit satisfactorily a wide range of experimental data in hippocampal and neocortical in vitro preparations. In particular, a model in which plasticity is driven by the postsynaptic calcium concentration was shown to reproduce successfully how synaptic changes depend on spike timing, specific spike patterns, and firing rate. The advantage of calcium-based rules is the possibility of predicting how changes in extracellular concentrations will affect plasticity. This is particularly significant in the view that in vitro studies are typically done at higher concentrations than the ones measured in vivo. Using such a rule, with parameters fitting in vitro data, we explore how long the memory of a particular synaptic change can be maintained in the presence of background neuronal activity, ubiquitously observed in cortex. We find that the memory time scales increase by several orders of magnitude when calcium concentrations are lowered from typical in vitro experiments to in vivo. Furthermore, we find that synaptic bistability further extends the memory time scale, and estimate that synaptic changes in vivo could be stable on the scale of weeks to months.

Bistability in the chemical master equation for dual phosphorylation cycles

Dual phospho/dephosphorylation cycles, as well as covalent enzymatic-catalyzed modifications of substrates are widely diffused within cellular systems and are crucial for the control of complex responses such as learning, memory, and cellular fate determination. Despite the large body of deterministic studies and the increasing work aimed at elucidating the effect of noise in such systems, some aspects remain unclear. Here we study the stationary distribution provided by the two-dimensional chemical master equation for a well-known model of a two step phospho/dephosphorylation cycle using the quasi-steady state approximation of enzymatic kinetics. Our aim is to analyze the role of fluctuations and the molecules distribution properties in the transition to a bistable regime. When detailed balance conditions are satisfied it is possible to compute equilibrium distributions in a closed and explicit form. When detailed balance is not satisfied, the stationary non-equilibrium state is strongly influenced by the chemical fluxes. In the last case, we show how the external field derived from the generation and recombination transition rates, can be decomposed by the Helmholtz theorem, into a conservative and a rotational (irreversible) part. Moreover, this decomposition allows to compute the stationary distribution via a perturbative approach. For a finite number of molecules there exists diffusion dynamics in a macroscopic region of the state space where a relevant transition rate between the two critical points is observed. Further, the stationary distribution function can be approximated by the solution of a Fokker-Planck equation. We illustrate the theoretical results using several numerical simulations.

Role of sensory input distribution and intrinsic connectivity in lateral amygdala during auditory fear conditioning: a computational study

We propose a novel reduced-order neuronal network modeling framework that includes an enhanced firing rate model and a corresponding synaptic calcium-based synaptic learning rule. Specifically, we propose enhancements to the Wilson-Cowan firing-rate neuron model that permit full spike-frequency adaptation seen in biological lateral amygdala (LA) neurons, while being sufficiently general to accommodate other spike-frequency patterns. We also report a technique to incorporate calcium-dependent plasticity in the synapses of the network using a regression scheme to link firing rate to postsynaptic calcium. Together, the single-cell model and the synaptic learning scheme constitute a general framework to develop computationally efficient neuronal networks that employ biologically realistic synaptic learning. The reduced-order modeling framework was validated using a previously reported biophysical conductance-based neuronal network model of a rodent LA that modeled features of Pavlovian conditioning and extinction of auditory fear (Li et al., 2009). The framework was then used to develop a larger LA network model to investigate the roles of tone and shock distributions and of intrinsic connectivity in auditory fear learning. The model suggested combinations of tone and shock densities that would provide experimental estimates of tone responsive and conditioned cell proportions. Furthermore, it provided several insights including how intrinsic connectivity might help distribute sensory inputs to produce conditioned responses in cells that do not directly receive both tone and shock inputs, and how a balance between potentiation of excitation and inhibition prevents stimulus generalization during fear learning.

The Mean Time to Express Synaptic Plasticity in Integrate-and-Express, Stochastic Models of Synaptic Plasticity Induction

Stochastic models of synaptic plasticity propose that single synapses perform a directed random walk of fixed step sizes in synaptic strength, thereby embracing the view that the mechanisms of synaptic plasticity constitute a stochastic dynamical system. However, fluctuations in synaptic strength present a formidable challenge to such an approach. We have previously proposed that single synapses must interpose an integration and filtering mechanism between the induction of synaptic plasticity and the expression of synaptic plasticity in order to control fluctuations. We analyze a class of three such mechanisms in the presence of possibly non-Markovian plasticity induction processes, deriving expressions for the mean expression time in these models. One of these filtering mechanisms constitutes a discrete low-pass filter that could be implemented on a small collection of molecules at single synapses, such as CaMKII, and we analyze this discrete filter in some detail. After considering Markov induction processes, we examine our own stochastic model of spike-timing-dependent plasticity, for which the probability density functions of the induction of plasticity steps have previously been derived. We determine the dependence of the mean time to express a plasticity step on pre- and postsynaptic firing rates in this model, and we also consider, numerically, the long-term stability against fluctuations of patterns of neuronal connectivity that typically emerge during neuronal development.

Stability against fluctuations: scaling, bifurcations, and spontaneous symmetry breaking in stochastic models of synaptic plasticity

In stochastic models of synaptic plasticity based on a random walk, the control of fluctuations is imperative. We have argued that synapses could act as low-pass filters, filtering plasticity induction steps before expressing a step change in synaptic strength. Earlier work showed, in simulation, that such a synaptic filter tames fluctuations very well, leading to patterns of synaptic connectivity that are stable for long periods of time. Here, we approach this problem analytically. We explicitly calculate the lifetime of meta-stable states of synaptic connectivity using a Fokker-Planck formalism in order to understand the dependence of this lifetime on both the plasticity step size and the filtering mechanism. We find that our analytical results agree very well with simulation results, despite having to make two approximations. Our analysis reveals, however, a deeper significance to the filtering mechanism and the plasticity step size. We show that a filter scales the step size into a smaller, effective step size. This scaling suggests that the step size may itself play the role of a temperature parameter, so that a filter cools the dynamics, thereby reducing the influence of fluctuations. Using the master equation, we explicitly demonstrate a bifurcation at a critical step size, confirming this interpretation. At this critical point, spontaneous symmetry breaking occurs in the class of stochastic models of synaptic plasticity that we consider.

Postsynaptic signal transduction models for long-term potentiation and depression

More than a hundred biochemical species, activated by neurotransmitters binding to transmembrane receptors, are important in long-term potentiation (LTP) and long-term depression (LTD). To investigate which species and interactions are critical for synaptic plasticity, many computational postsynaptic signal transduction models have been developed. The models range from simple models with a single reversible reaction to detailed models with several hundred kinetic reactions. In this study, more than a hundred models are reviewed, and their features are compared and contrasted so that similarities and differences are more readily apparent. The models are classified according to the type of synaptic plasticity that is modeled (LTP or LTD) and whether they include diffusion or electrophysiological phenomena. Other characteristics that discriminate the models include the phase of synaptic plasticity modeled (induction, expression, or maintenance) and the simulation method used (deterministic or stochastic). We find that models are becoming increasingly sophisticated, by including stochastic properties, integrating with electrophysiological properties of entire neurons, or incorporating diffusion of signaling molecules. Simpler models continue to be developed because they are computationally efficient and allow theoretical analysis. The more complex models permit investigation of mechanisms underlying specific properties and experimental verification of model predictions. Nonetheless, it is difficult to fully comprehend the evolution of these models because (1) several models are not described in detail in the publications, (2) only a few models are provided in existing model databases, and (3) comparison to previous models is lacking. We conclude that the value of these models for understanding molecular mechanisms of synaptic plasticity is increasing and will be enhanced further with more complete descriptions and sharing of the published models.

Interplay of the magnitude and time-course of postsynaptic Ca2+ concentration in producing spike timing-dependent plasticity

Synaptic strength can be modified by the relative timing of pre- and postsynaptic activity, a phenomenon termed spike timing-dependent plasticity (STDP). Studies of neurons in the hippocampus and in other regions have found that when presynaptic activity occurs within a narrow time window, typically 10 or 20 ms, before postsynaptic activity, long-term potentiation (LTP) is induced, while if presynaptic activity occurs within a similar time window after postsynaptic activity, long-term depression (LTD) results. The mechanisms underlying these modifications are not completely understood, although there is strong evidence that the postsynaptic Ca (2+) concentration plays a central role. Some previous modeling of STDP has focused on the dynamics of the postsynaptic Ca (2+) concentration, while other work has studied biophysical mechanisms of how a synapse can exist in, and switch between, different states corresponding to LTP and LTD. Building on previous work in these two areas we have developed the first low level STDP model of a tristable biochemical system that incorporates induction and maintenance of both LTP and LTD. Our model is able to explain the STDP observed in hippocampal neurons in response to pre- and postsynaptic pulse pairs, using only parameters derived from previous work and without the need for parameter fine-tuning. Our results also give insight into how and why the time course of the postsynaptic Ca (2+) concentration can lead to either LTP or LTD, and suggest that voltage dependent calcium channels play a key role.

Mechanisms of induction and maintenance of spike-timing dependent plasticity in biophysical synapse models

We review biophysical models of synaptic plasticity, with a focus on spike-timing dependent plasticity (STDP). The common property of the discussed models is that synaptic changes depend on the dynamics of the intracellular calcium concentration, which itself depends on pre- and postsynaptic activity. We start by discussing simple models in which plasticity changes are based directly on calcium amplitude and dynamics. We then consider models in which dynamic intracellular signaling cascades form the link between the calcium dynamics and the plasticity changes. Both mechanisms of induction of STDP (through the ability of pre/postsynaptic spikes to evoke changes in the state of the synapse) and of maintenance of the evoked changes (through bistability) are discussed.

A Ca-Based Computational Model for NMDA Receptor-Dependent Synaptic Plasticity at Individual Post-Synaptic Spines in the Hippocampus

Associative synaptic plasticity is synapse specific and requires coincident activity in pre-synaptic and post-synaptic neurons to activate NMDA receptors (NMDARs). The resultant Ca²⁺ influx is the critical trigger for the induction of synaptic plasticity. Given its centrality for the induction of synaptic plasticity, a model for NMDAR activation incorporating the timing of pre-synaptic glutamate release and post-synaptic depolarization by back-propagating action potentials could potentially predict the pre- and post-synaptic spike patterns required to induce synaptic plasticity. We have developed such a model by incorporating currently available data on the timecourse and amplitude of the post-synaptic membrane potential within individual spines. We couple this with data on the kinetics of synaptic NMDARs and then use the model to predict the continuous spine [Ca²⁺] in response to regular or irregular pre- and post-synaptic spike patterns. We then incorporate experimental data from synaptic plasticity induction protocols by regular activity patterns to couple the predicted local peak [Ca²⁺] to changes in synaptic strength. We find that our model accurately describes [Ca²⁺] in dendritic spines resulting from NMDAR activation during pre-synaptic and post-synaptic activity when compared to previous experimental observations. The model also replicates the experimentally determined plasticity outcome of regular and irregular spike patterns when applied to a single synapse. This model could therefore be used to predict the induction of synaptic plasticity under a variety of experimental conditions and spike patterns.

Temperature control of fimbriation circuit switch in uropathogenic Escherichia coli: quantitative analysis via automated model abstraction

Uropathogenic Escherichia coli (UPEC) represent the predominant cause of urinary tract infections (UTIs). A key UPEC molecular virulence mechanism is type 1 fimbriae, whose expression is controlled by the orientation of an invertible chromosomal DNA element-the fim switch. Temperature has been shown to act as a major regulator of fim switching behavior and is overall an important indicator as well as functional feature of many urologic diseases, including UPEC host-pathogen interaction dynamics. Given this panoptic physiological role of temperature during UTI progression and notable empirical challenges to its direct in vivo studies, in silico modeling of corresponding biochemical and biophysical mechanisms essential to UPEC pathogenicity may significantly aid our understanding of the underlying disease processes. However, rigorous computational analysis of biological systems, such as fim switch temperature control circuit, has hereto presented a notoriously demanding problem due to both the substantial complexity of the gene regulatory networks involved as well as their often characteristically discrete and stochastic dynamics. To address these issues, we have developed an approach that enables automated multiscale abstraction of biological system descriptions based on reaction kinetics. Implemented as a computational tool, this method has allowed us to efficiently analyze the modular organization and behavior of the E. coli fimbriation switch circuit at different temperature settings, thus facilitating new insights into this mode of UPEC molecular virulence regulation. In particular, our results suggest that, with respect to its role in shutting down fimbriae expression, the primary function of FimB recombinase may be to effect a controlled down-regulation (rather than increase) of the ON-to-OFF fim switching rate via temperature-dependent suppression of competing dynamics mediated by recombinase FimE. Our computational analysis further implies that this down-regulation mechanism could be particularly significant inside the host environment, thus potentially contributing further understanding toward the development of novel therapeutic approaches to UPEC-caused UTIs.

A kinetic model of dopamine- and calcium-dependent striatal synaptic plasticity

Corticostriatal synapse plasticity of medium spiny neurons is regulated by glutamate input from the cortex and dopamine input from the substantia nigra. While cortical stimulation alone results in long-term depression (LTD), the combination with dopamine switches LTD to long-term potentiation (LTP), which is known as dopamine-dependent plasticity. LTP is also induced by cortical stimulation in magnesium-free solution, which leads to massive calcium influx through NMDA-type receptors and is regarded as calcium-dependent plasticity. Signaling cascades in the corticostriatal spines are currently under investigation. However, because of the existence of multiple excitatory and inhibitory pathways with loops, the mechanisms regulating the two types of plasticity remain poorly understood. A signaling pathway model of spines that express D1-type dopamine receptors was constructed to analyze the dynamic mechanisms of dopamine- and calcium-dependent plasticity. The model incorporated all major signaling molecules, including dopamine- and cyclic AMP-regulated phosphoprotein with a molecular weight of 32 kDa (DARPP32), as well as AMPA receptor trafficking in the post-synaptic membrane. Simulations with dopamine and calcium inputs reproduced dopamine- and calcium-dependent plasticity. Further in silico experiments revealed that the positive feedback loop consisted of protein kinase A (PKA), protein phosphatase 2A (PP2A), and the phosphorylation site at threonine 75 of DARPP-32 (Thr75) served as the major switch for inducing LTD and LTP. Calcium input modulated this loop through the PP2B (phosphatase 2B)-CK1 (casein kinase 1)-Cdk5 (cyclin-dependent kinase 5)-Thr75 pathway and PP2A, whereas calcium and dopamine input activated the loop via PKA activation by cyclic AMP (cAMP). The positive feedback loop displayed robust bi-stable responses following changes in the reaction parameters. Increased basal dopamine levels disrupted this dopamine-dependent plasticity. The present model elucidated the mechanisms involved in bidirectional regulation of corticostriatal synapses and will allow for further exploration into causes and therapies for dysfunctions such as drug addiction.

Recent brain imaging and neurophysiological studies suggest that the striatum, the start of the basal ganglia circuit, plays a major role in value-based decision making and behavioral disorders such as drug addiction. The plasticity of synaptic input from the cerebral cortex to output neurons of the striatum, which are medium spiny neurons, depends on interactions between glutamate input from the cortex and dopaminergic input from the midbrain. It also links sensory and cognitive states in the cortex with reward-oriented action outputs. The mechanisms involved in molecular cascades that transmit glutamate and dopamine inputs to changes in postsynaptic glutamate receptors are very complex and it is difficult to intuitively understand the mechanism. Therefore, a biochemical network model was constructed, and computer simulations were performed. The model reproduced dopamine-dependent and calcium-dependent forms of long-term depression (LTD) and potentiation (LTP) of corticostriatal synapses. Further in silico experiments revealed that a positive feedback loop formed by proteins, the protein specifically expressed in the striatum, served as the major switch for inducing LTD and LTP. This model could allow us to understand dynamic constraints in reward-dependent learning, as well as causes and therapies of dopamine-related disorders such as drug addiction.