Synaptic Information Storage Capacity Measured With Information Theory.

Variation in the strength of synapses can be quantified by measuring the anatomical properties of synapses. Quantifying precision of synaptic plasticity is fundamental to understanding information storage and retrieval in neural circuits. Synapses from the same axon onto the same dendrite have a common history of coactivation, making them ideal candidates for determining the precision of synaptic plasticity based on the similarity of their physical dimensions. Here, the precision and amount of information stored in synapse dimensions were quantified with Shannon information theory, expanding prior analysis that used signal detection theory (Bartol et al., 2015). The two methods were compared using dendritic spine head volumes in the middle of the stratum radiatum of hippocampal area CA1 as well-defined measures of synaptic strength. Information theory delineated the number of distinguishable synaptic strengths based on nonoverlapping bins of dendritic spine head volumes. Shannon entropy was applied to measure synaptic information storage capacity (SISC) and resulted in a lower bound of 4.1 bits and upper bound of 4.59 bits of information based on 24 distinguishable sizes. We further compared the distribution of distinguishable sizes and a uniform distribution using Kullback-Leibler divergence and discovered that there was a nearly uniform distribution of spine head volumes across the sizes, suggesting optimal use of the distinguishable values. Thus, SISC provides a new analytical measure that can be generalized to probe synaptic strengths and capacity for plasticity in different brain regions of different species and among animals raised in different conditions or during learning. How brain diseases and disorders affect the precision of synaptic plasticity can also be probed.

MCell4 with BioNetGen: A Monte Carlo simulator of rule-based reaction-diffusion systems with Python interface.

Biochemical signaling pathways in living cells are often highly organized into spatially segregated volumes, membranes, scaffolds, subcellular compartments, and organelles comprising small numbers of interacting molecules. At this level of granularity stochastic behavior dominates, well-mixed continuum approximations based on concentrations break down and a particle-based approach is more accurate and more efficient. We describe and validate a new version of the open-source MCell simulation program (MCell4), which supports generalized 3D Monte Carlo modeling of diffusion and chemical reaction of discrete molecules and macromolecular complexes in solution, on surfaces representing membranes, and combinations thereof. The main improvements in MCell4 compared to the previous versions, MCell3 and MCell3-R, include a Python interface and native BioNetGen reaction language (BNGL) support. MCell4's Python interface opens up completely new possibilities for interfacing with external simulators to allow creation of sophisticated event-driven multiscale/multiphysics simulations. The native BNGL support, implemented through a new open-source library libBNG (also introduced in this paper), provides the capability to run a given BNGL model spatially resolved in MCell4 and, with appropriate simplifying assumptions, also in the BioNetGen simulation environment, greatly accelerating and simplifying model validation and comparison.

A spatial model of autophosphorylation of CaMKII in a glutamatergic spine suggests a network-driven kinetic mechanism for bistable changes in synaptic strength.

Activation of N-methyl-D-aspartate-type glutamate receptors (NMDARs) at synapses in the CNS triggers changes in synaptic strength that underlie memory formation in response to strong synaptic stimuli. The primary target of Ca2+ flowing through NMDARs is Ca2+/calmodulin-dependent protein kinase II (CaMKII) which forms dodecameric holoenzymes that are highly concentrated at the postsynaptic site. Activation of CaMKII is necessary to trigger long-term potentiation of synaptic strength (LTP), and is prolonged by autophosphorylation of subunits within the holoenzyme. Here we use MCell4, an agent-based, stochastic, modeling platform to model CaMKII holoenzymes placed within a realistic spine geometry. We show how two mechanisms of regulation of CaMKII, 'Ca2+-calmodulin-trapping (CaM-trapping)' and dephosphorylation by protein phosphatase-1 (PP1) shape the autophosphorylation response during a repeated high-frequency stimulus. Our simulation results suggest that autophosphorylation of CaMKII does not constitute a bistable switch. Instead, prolonged but temporary, autophosphorylation of CaMKII may contribute to a biochemical-network-based 'kinetic proof-reading" mechanism that controls induction of synaptic plasticity.

Long-Term Potentiation Produces a Sustained Expansion of Synaptic Information Storage Capacity in Adult Rat Hippocampus.

Long-term potentiation (LTP) has become a standard model for investigating synaptic mechanisms of learning and memory. Increasingly, it is of interest to understand how LTP affects the synaptic information storage capacity of the targeted population of synapses. Here, structural synaptic plasticity during LTP was explored using three-dimensional reconstruction from serial section electron microscopy. Storage capacity was assessed by applying a new analytical approach, Shannon information theory, to delineate the number of functionally distinguishable synaptic strengths. LTP was induced by delta-burst stimulation of perforant pathway inputs to the middle molecular layer of hippocampal dentate granule cells in adult rats. Spine head volumes were measured as predictors of synaptic strength and compared between LTP and control hemispheres at 30 min and 2 hr after the induction of LTP. Synapses from the same axon onto the same dendrite were used to determine the precision of synaptic plasticity based on the similarity of their physical dimensions. Shannon entropy was measured by exploiting the frequency of spine heads in functionally distinguishable sizes to assess the degree to which LTP altered the number of bits of information storage. Outcomes from these analyses reveal that LTP expanded storage capacity; the distribution of spine head volumes was increased from 2 bits in controls to 3 bits at 30 min and 2.7 bits at 2 hr after the induction of LTP. Furthermore, the distribution of spine head volumes was more uniform across the increased number of functionally distinguishable sizes following LTP, thus achieving more efficient use of coding space across the population of synapses.

Computer simulations predict the impact of neuronal atrophy on the calcium dynamics in Huntington's disease.

One of the early hallmarks of Huntington's disease (HD) is neuronal cell atrophy, especially in the striatum, underlying motor dysfunction in HD. Here using a computer model, we have predicted the impact of cell shrinkage on calcium dynamics at the cellular level. Our model indicates that as cytosolic volume decreases, the amplitude of calcium transients increases and the endoplasmic reticulum (ER) becomes more leaky due to calcium-induced calcium release and a "toxic" positive feedback mechanism mediated by ryanodine receptors that greatly increases calcium release into the cytosol. The excessive calcium release from ER saturates the calcium buffering capacity of calbindin and forces further accumulation of free calcium in the cytosol and cellular compartments including mitochondria. This leads to imbalance of calcium in both cytosol and ER regions. Excessive calcium accumulation in the cytosol can damage the mitochondria resulting in metabolic dysfunction in the cell consistent with the pathology of HD. Our computational model points toward potential drug targets and can accelerate and greatly help the experimental studies of HD paving the way for treatments of patients suffering from HD.

Mitochondrial morphology governs ATP production rate.

Life is based on energy conversion. In particular, in the nervous system, significant amounts of energy are needed to maintain synaptic transmission and homeostasis. To a large extent, neurons depend on oxidative phosphorylation in mitochondria to meet their high energy demand. For a comprehensive understanding of the metabolic demands in neuronal signaling, accurate models of ATP production in mitochondria are required. Here, we present a thermodynamically consistent model of ATP production in mitochondria based on previous work. The significant improvement of the model is that the reaction rate constants are set such that detailed balance is satisfied. Moreover, using thermodynamic considerations, the dependence of the reaction rate constants on membrane potential, pH, and substrate concentrations are explicitly provided. These constraints assure that the model is physically plausible. Furthermore, we explore different parameter regimes to understand in which conditions ATP production or its export are the limiting steps in making ATP available in the cytosol. The outcomes reveal that, under the conditions used in our simulations, ATP production is the limiting step and not its export. Finally, we performed spatial simulations with nine 3-D realistic mitochondrial reconstructions and linked the ATP production rate in the cytosol with morphological features of the organelles.

Multiscale modeling of presynaptic dynamics from molecular to mesoscale.

Chemical synapses exhibit a diverse array of internal mechanisms that affect the dynamics of transmission efficacy. Many of these processes, such as release of neurotransmitter and vesicle recycling, depend strongly on activity-dependent influx and accumulation of Ca2+. To model how each of these processes may affect the processing of information in neural circuits, and how their dysfunction may lead to disease states, requires a computationally efficient modelling framework, capable of generating accurate phenomenology without incurring a heavy computational cost per synapse. Constructing a phenomenologically realistic model requires the precise characterization of the timing and probability of neurotransmitter release. Difficulties arise in that functional forms of instantaneous release rate can be difficult to extract from noisy data without running many thousands of trials, and in biophysical synapses, facilitation of per-vesicle release probability is confounded by depletion. To overcome this, we obtained traces of free Ca2+ concentration in response to various action potential stimulus trains from a molecular MCell model of a hippocampal Schaffer collateral axon. Ca2+ sensors were placed at varying distance from a voltage-dependent calcium channel (VDCC) cluster, and Ca2+ was buffered by calbindin. Then, using the calcium traces to drive deterministic state vector models of synaptotagmin 1 and 7 (Syt-1/7), which respectively mediate synchronous and asynchronous release in excitatory hippocampal synapses, we obtained high-resolution profiles of instantaneous release rate, to which we applied functional fits. Synchronous vesicle release occurred predominantly within half a micron of the source of spike-evoked Ca2+ influx, while asynchronous release occurred more consistently at all distances. Both fast and slow mechanisms exhibited multi-exponential release rate curves, whose magnitudes decayed exponentially with distance from the Ca2+ source. Profile parameters facilitate on different time scales according to a single, general facilitation function. These functional descriptions lay the groundwork for efficient mesoscale modelling of vesicular release dynamics.

Morphological principles of neuronal mitochondria.

In the highly dynamic metabolic landscape of a neuron, mitochondrial membrane architectures can provide critical insight into the unique energy balance of the cell. Current theoretical calculations of functional outputs like adenosine triphosphate and heat often represent mitochondria as idealized geometries, and therefore, can miscalculate the metabolic fluxes. To analyze mitochondrial morphology in neurons of mouse cerebellum neuropil, 3D tracings of complete synaptic and axonal mitochondria were constructed using a database of serial transmission electron microscopy (TEM) tomography images and converted to watertight meshes with minimal distortion of the original microscopy volumes with a granularity of 1.64 nanometer isotropic voxels. The resulting in-silico representations were subsequently quantified by differential geometry methods in terms of the mean and Gaussian curvatures, surface areas, volumes, and membrane motifs, all of which can alter the metabolic output of the organelle. Finally, we identify structural motifs present across this population of mitochondria, which may contribute to future modeling studies of mitochondrial physiology and metabolism in neurons.

Markov Chain Abstractions of Electrochemical Reaction-Diffusion in Synaptic Transmission for Neuromorphic Computing.

Progress in computational neuroscience toward understanding brain function is challenged both by the complexity of molecular-scale electrochemical interactions at the level of individual neurons and synapses and the dimensionality of network dynamics across the brain covering a vast range of spatial and temporal scales. Our work abstracts an existing highly detailed, biophysically realistic 3D reaction-diffusion model of a chemical synapse to a compact internal state space representation that maps onto parallel neuromorphic hardware for efficient emulation at a very large scale and offers near-equivalence in input-output dynamics while preserving biologically interpretable tunable parameters.

NMDAR-dependent long-term depression is associated with increased short term plasticity through autophagy mediated loss of PSD-95.

Long-term depression (LTD) of synaptic strength can take multiple forms and contribute to circuit remodeling, memory encoding or erasure. The generic term LTD encompasses various induction pathways, including activation of NMDA, mGlu or P2X receptors. However, the associated specific molecular mechanisms and effects on synaptic physiology are still unclear. We here compare how NMDAR- or P2XR-dependent LTD affect synaptic nanoscale organization and function in rodents. While both LTDs are associated with a loss and reorganization of synaptic AMPARs, only NMDAR-dependent LTD induction triggers a profound reorganization of PSD-95. This modification, which requires the autophagy machinery to remove the T19-phosphorylated form of PSD-95 from synapses, leads to an increase in AMPAR surface mobility. We demonstrate that these post-synaptic changes that occur specifically during NMDAR-dependent LTD result in an increased short-term plasticity improving neuronal responsiveness of depressed synapses. Our results establish that P2XR- and NMDAR-mediated LTD are associated to functionally distinct forms of LTD.

A Discrete Presynaptic Vesicle Cycle for Neuromodulator Receptors.

A major function of GPCRs is to inhibit presynaptic neurotransmitter release, requiring ligand-activated receptors to couple locally to effectors at terminals. The current understanding of how this is achieved is through receptor immobilization on the terminal surface. Here, we show that opioid peptide receptors, GPCRs that mediate highly sensitive presynaptic inhibition, are instead dynamic in axons. Opioid receptors diffuse rapidly throughout the axon surface and internalize after ligand-induced activation specifically at presynaptic terminals. We delineate a parallel regulated endocytic cycle for GPCRs operating at the presynapse, separately from the synaptic vesicle cycle, which clears activated receptors from the surface of terminals and locally reinserts them to maintain the diffusible surface pool. We propose an alternate strategy for achieving local control of presynaptic effectors that, opposite to using receptor immobilization and enforced proximity, is based on lateral mobility of receptors and leverages the inherent allostery of GPCR-effector coupling.

A multi-state model of the CaMKII dodecamer suggests a role for calmodulin in maintenance of autophosphorylation.

Ca2+/calmodulin-dependent protein kinase II (CaMKII) accounts for up to 2 percent of all brain protein and is essential to memory function. CaMKII activity is known to regulate dynamic shifts in the size and signaling strength of neuronal connections, a process known as synaptic plasticity. Increasingly, computational models are used to explore synaptic plasticity and the mechanisms regulating CaMKII activity. Conventional modeling approaches may exclude biophysical detail due to the impractical number of state combinations that arise when explicitly monitoring the conformational changes, ligand binding, and phosphorylation events that occur on each of the CaMKII holoenzyme's subunits. To manage the combinatorial explosion without necessitating bias or loss in biological accuracy, we use a specialized syntax in the software MCell to create a rule-based model of a twelve-subunit CaMKII holoenzyme. Here we validate the rule-based model against previous experimental measures of CaMKII activity and investigate molecular mechanisms of CaMKII regulation. Specifically, we explore how Ca2+/CaM-binding may both stabilize CaMKII subunit activation and regulate maintenance of CaMKII autophosphorylation. Noting that Ca2+/CaM and protein phosphatases bind CaMKII at nearby or overlapping sites, we compare model scenarios in which Ca2+/CaM and protein phosphatase do or do not structurally exclude each other's binding to CaMKII. Our results suggest a functional mechanism for the so-called "CaM trapping" phenomenon, wherein Ca2+/CaM may structurally exclude phosphatase binding and thereby prolong CaMKII autophosphorylation. We conclude that structural protection of autophosphorylated CaMKII by Ca2+/CaM may be an important mechanism for regulation of synaptic plasticity.

Mitochondrial morphology provides a mechanism for energy buffering at synapses.

Mitochondria as the main energy suppliers of eukaryotic cells are highly dynamic organelles that fuse, divide and are transported along the cytoskeleton to ensure cellular energy homeostasis. While these processes are well established, substantial evidence indicates that the internal structure is also highly variable in dependence on metabolic conditions. However, a quantitative mechanistic understanding of how mitochondrial morphology affects energetic states is still elusive. To address this question, we here present an agent-based multiscale model that integrates three-dimensional morphologies from electron microscopy tomography with the molecular dynamics of the main ATP producing components. We apply our modeling approach to mitochondria at the synapse which is the largest energy consumer within the brain. Interestingly, comparing the spatiotemporal simulations with a corresponding space-independent approach, we find minor spatial effects when the system relaxes toward equilibrium but a qualitative difference in fluctuating environments. These results suggest that internal mitochondrial morphology is not only optimized for ATP production but also provides a mechanism for energy buffering and may represent a mechanism for cellular robustness.

Geometric principles of second messenger dynamics in dendritic spines.

Dendritic spines are small, bulbous protrusions along dendrites in neurons and play a critical role in synaptic transmission. Dendritic spines come in a variety of shapes that depend on their developmental state. Additionally, roughly 14-19% of mature spines have a specialized endoplasmic reticulum called the spine apparatus. How does the shape of a postsynaptic spine and its internal organization affect the spatio-temporal dynamics of short timescale signaling? Answers to this question are central to our understanding the initiation of synaptic transmission, learning, and memory formation. In this work, we investigated the effect of spine and spine apparatus size and shape on the spatio-temporal dynamics of second messengers using mathematical modeling using reaction-diffusion equations in idealized geometries (ellipsoids, spheres, and mushroom-shaped). Our analyses and simulations showed that in the short timescale, spine size and shape coupled with the spine apparatus geometries govern the spatiotemporal dynamics of second messengers. We show that the curvature of the geometries gives rise to pseudo-harmonic functions, which predict the locations of maximum and minimum concentrations along the spine head. Furthermore, we showed that the lifetime of the concentration gradient can be fine-tuned by localization of fluxes on the spine head and varying the relative curvatures and distances between the spine apparatus and the spine head. Thus, we have identified several key geometric determinants of how the spine head and spine apparatus may regulate the short timescale chemical dynamics of small molecules that control synaptic plasticity.

Learning moment closure in reaction-diffusion systems with spatial dynamic Boltzmann distributions.

Many physical systems are described by probability distributions that evolve in both time and space. Modeling these systems is often challenging due to their large state space and analytically intractable or computationally expensive dynamics. To address these problems, we study a machine-learning approach to model reduction based on the Boltzmann machine. Given the form of the reduced model Boltzmann distribution, we introduce an autonomous differential equation system for the interactions appearing in the energy function. The reduced model can treat systems in continuous space (described by continuous random variables), for which we formulate a variational learning problem using the adjoint method to determine the right-hand sides of the differential equations. This approach can be used to enforce a reduced physical model by a suitable parametrization of the differential equations. The parametrization we employ uses the basis functions from finite-element methods, which can be used to model any physical system. One application domain for such physics-informed learning algorithms is to modeling reaction-diffusion systems. We study a lattice version of the Rössler chaotic oscillator, which illustrates the accuracy of the moment closure approximation made by the method and its dimensionality reduction power.

MCell-R: A Particle-Resolution Network-Free Spatial Modeling Framework.

Spatial heterogeneity can have dramatic effects on the biochemical networks that drive cell regulation and decision-making. For this reason, a number of methods have been developed to model spatial heterogeneity and incorporated into widely used modeling platforms. Unfortunately, the standard approaches for specifying and simulating chemical reaction networks become untenable when dealing with multistate, multicomponent systems that are characterized by combinatorial complexity. To address this issue, we developed MCell-R, a framework that extends the particle-based spatial Monte Carlo simulator, MCell, with the rule-based model specification and simulation capabilities provided by BioNetGen and NFsim. The BioNetGen syntax enables the specification of biomolecules as structured objects whose components can have different internal states that represent such features as covalent modification and conformation and which can bind components of other molecules to form molecular complexes. The network-free simulation algorithm used by NFsim enables efficient simulation of rule-based models even when the size of the network implied by the biochemical rules is too large to enumerate explicitly, which frequently occurs in detailed models of biochemical signaling. The result is a framework that can efficiently simulate systems characterized by combinatorial complexity at the level of spatially resolved individual molecules over biologically relevant time and length scales.

Pre-post synaptic alignment through neuroligin-1 tunes synaptic transmission efficiency.

The nanoscale organization of neurotransmitter receptors regarding pre-synaptic release sites is a fundamental determinant of the synaptic transmission amplitude and reliability. How modifications in the pre- and post-synaptic machinery alignments affects synaptic currents, has only been addressed with computer modelling. Using single molecule super-resolution microscopy, we found a strong spatial correlation between AMPA receptor (AMPAR) nanodomains and the post-synaptic adhesion protein neuroligin-1 (NLG1). Expression of a truncated form of NLG1 disrupted this correlation without affecting the intrinsic AMPAR organization, shifting the pre-synaptic release machinery away from AMPAR nanodomains. Electrophysiology in dissociated and organotypic hippocampal rodent cultures shows these treatments significantly decrease AMPAR-mediated miniature and EPSC amplitudes. Computer modelling predicts that ~100 nm lateral shift between AMPAR nanoclusters and glutamate release sites induces a significant reduction in AMPAR-mediated currents. Thus, our results suggest the synapses necessity to release glutamate precisely in front of AMPAR nanodomains, to maintain a high synaptic responses efficiency.

Efficient models of polymerization applied to FtsZ ring assembly in Escherichia coli.

High protein concentrations complicate modeling of polymer assembly kinetics by introducing structural complexity and a large variety of protein forms. We present a modeling approach that achieves orders of magnitude speed-up by replacing distributions of lengths and widths with their average counterparts and by introducing a hierarchical classification of species and reactions into sets. We have used this model to study FtsZ ring assembly in Escherichia coli The model's prediction of key features of the ring formation, such as time to reach the steady state, total concentration of FtsZ species in the ring, total concentration of monomers, and average dimensions of filaments and bundles, are all in agreement with the experimentally observed values. Besides validating our model against the in vivo observations, this study fills some knowledge gaps by proposing a specific structure of the ring, describing the influence of the total concentration in short and long kinetics processes, determining some characteristic mechanisms in polymer assembly regulation, and providing insights about the role of ZapA proteins, critical components for both positioning and stability of the ring.

Long-term potentiation expands information content of hippocampal dentate gyrus synapses.

An approach combining signal detection theory and precise 3D reconstructions from serial section electron microscopy (3DEM) was used to investigate synaptic plasticity and information storage capacity at medial perforant path synapses in adult hippocampal dentate gyrus in vivo. Induction of long-term potentiation (LTP) markedly increased the frequencies of both small and large spines measured 30 minutes later. This bidirectional expansion resulted in heterosynaptic counterbalancing of total synaptic area per unit length of granule cell dendrite. Control hemispheres exhibited 6.5 distinct spine sizes for 2.7 bits of storage capacity while LTP resulted in 12.9 distinct spine sizes (3.7 bits). In contrast, control hippocampal CA1 synapses exhibited 4.7 bits with much greater synaptic precision than either control or potentiated dentate gyrus synapses. Thus, synaptic plasticity altered total capacity, yet hippocampal subregions differed dramatically in their synaptic information storage capacity, reflecting their diverse functions and activation histories.