A minimal presynaptic protein machinery mediating synchronous and asynchronous exocytosis and short-term plasticity.

Neurotransmitters are released from synaptic vesicles with remarkable precision in response to presynaptic Ca2+ influx but exhibit significant heterogeneity in exocytosis timing and efficacy based on the recent history of activity. This heterogeneity is critical for information transfer in the brain, yet its molecular basis remains poorly understood. Here, we employ a biochemically-defined fusion assay under physiologically-relevant conditions to delineate the minimal protein machinery sufficient to account for different modes of Ca2+-triggered vesicle fusion and short-term facilitation. We find that Synaptotagmin-1, Synaptotagmin-7, and Complexin, synergistically restrain SNARE complex assembly, thus preserving vesicles in a stably docked state at rest. Upon Ca2+ activation, Synaptotagmin-1 induces rapid vesicle fusion, while Synaptotagmin-7 mediates delayed fusion. Competitive binding of Synaptotagmin-1 and Synaptotagmin-7 to the same SNAREs, coupled with differential rates of Ca2+-triggered fusion clamp reversal, govern the kinetics of vesicular fusion. Under conditions mimicking sustained neuronal activity, the Synaptotagmin-7 fusion clamp is destabilized by the elevated basal Ca2+ concentration, thereby enhancing the synchronous component of fusion. These findings provide a direct demonstration that a small set of proteins is sufficient to account for how nerve terminals adapt and regulate the Ca2+-evoked neurotransmitter exocytosis process to support their specialized functions in the nervous system.

An exploratory computational analysis in mice brain networks of widespread epileptic seizure onset locations along with potential strategies for effective intervention and propagation control.

Mean-field models have been developed to replicate key features of epileptic seizure dynamics. However, the precise mechanisms and the role of the brain area responsible for seizure onset and propagation remain incompletely understood. In this study, we employ computational methods within The Virtual Brain framework and the Epileptor model to explore how the location and connectivity of an Epileptogenic Zone (EZ) in a mouse brain are related to focal seizures (seizures that start in one brain area and may or may not remain localized), with a specific focus on the hippocampal region known for its association with epileptic seizures. We then devise computational strategies to confine seizures (prevent widespread propagation), simulating medical-like treatments such as tissue resection and the application of an anti-seizure drugs or neurostimulation to suppress hyperexcitability. Through selectively removing (blocking) specific connections informed by the structural connectome and graph network measurements or by locally reducing outgoing connection weights of EZ areas, we demonstrate that seizures can be kept constrained around the EZ region. We successfully identified the minimal connections necessary to prevent widespread seizures, with a particular focus on minimizing surgical or medical intervention while simultaneously preserving the original structural connectivity and maximizing brain functionality.

The release of inhibition model reproduces kinetics and plasticity of neurotransmitter release in central synapses.

Calcium-evoked release of neurotransmitters from synaptic vesicles (SVs) is catalysed by SNARE proteins. The predominant view is that, at rest, complete assembly of SNARE complexes is inhibited ('clamped') by synaptotagmin and complexin molecules. Calcium binding by synaptotagmins releases this fusion clamp and triggers fast SV exocytosis. However, this model has not been quantitatively tested over physiological timescales. Here we describe an experimentally constrained computational modelling framework to quantitatively assess how the molecular architecture of the fusion clamp affects SV exocytosis. Our results argue that the 'release-of-inhibition' model can indeed account for fast calcium-activated SV fusion, and that dual binding of synaptotagmin-1 and synaptotagmin-7 to the same SNARE complex enables synergistic regulation of the kinetics and plasticity of neurotransmitter release. The developed framework provides a powerful and adaptable tool to link the molecular biochemistry of presynaptic proteins to physiological data and efficiently test the plausibility of calcium-activated neurotransmitter release models.

The release of inhibition model reproduces kinetics and plasticity of neurotransmitter release in central synapses.

Calcium-evoked release of neurotransmitters from synaptic vesicles (SVs) is catalysed by SNARE proteins. The predominant view is that, at rest, complete assembly of SNARE complexes is inhibited ('clamped') by synaptotagmin and complexin molecules. Calcium binding by synaptotagmins releases this fusion clamp and triggers fast SV exocytosis. However, this model has not been quantitatively tested over physiological timescales. Here we describe an experimentally constrained computational modelling framework to quantitatively assess how the molecular architecture of the fusion clamp affects SV exocytosis. Our results argue that the "release-of-inhibition" model can indeed account for fast calcium-activated SV fusion, and that dual binding of synaptotagmin-1 and synaptotagmin-7 to the same SNARE complex enables synergistic regulation of the kinetics and plasticity of neurotransmitter release. The developed framework provides a powerful and adaptable tool to link the molecular biochemistry of presynaptic proteins to physiological data and efficiently test the plausibility of calcium-activated neurotransmitter release models.

Asynchronous glutamate release is enhanced in low release efficacy synapses and dispersed across the active zone.

The balance between fast synchronous and delayed asynchronous release of neurotransmitters has a major role in defining computational properties of neuronal synapses and regulation of neuronal network activity. However, how it is tuned at the single synapse level remains poorly understood. Here, using the fluorescent glutamate sensor SF-iGluSnFR, we image quantal vesicular release in tens to hundreds of individual synaptic outputs from single pyramidal cells with 4 millisecond temporal and 75 nm spatial resolution. We find that the ratio between synchronous and asynchronous synaptic vesicle exocytosis varies extensively among synapses supplied by the same axon, and that the synchronicity of release is reduced at low release probability synapses. We further demonstrate that asynchronous exocytosis sites are more widely distributed within the release area than synchronous sites. Together, our results reveal a universal relationship between the two major functional properties of synapses - the timing and the overall efficacy of neurotransmitter release.

Two opposite voltage-dependent currents control the unusual early development pattern of embryonic Renshaw cell electrical activity.

Renshaw cells (V1R) are excitable as soon as they reach their final location next to the spinal motoneurons and are functionally heterogeneous. Using multiple experimental approaches, in combination with biophysical modeling and dynamical systems theory, we analyzed, for the first time, the mechanisms underlying the electrophysiological properties of V1R during early embryonic development of the mouse spinal cord locomotor networks (E11.5-E16.5). We found that these interneurons are subdivided into several functional clusters from E11.5 and then display an unexpected transitory involution process during which they lose their ability to sustain tonic firing. We demonstrated that the essential factor controlling the diversity of the discharge pattern of embryonic V1R is the ratio of a persistent sodium conductance to a delayed rectifier potassium conductance. Taken together, our results reveal how a simple mechanism, based on the synergy of two voltage-dependent conductances that are ubiquitous in neurons, can produce functional diversity in embryonic V1R and control their early developmental trajectory.

Fluorescence lifetime imaging reveals regulation of presynaptic Ca2+ by glutamate uptake and mGluRs, but not somatic voltage in cortical neurons.

Brain function relies on vesicular release of neurotransmitters at chemical synapses. The release probability depends on action potential-evoked presynaptic Ca2+ entry, but also on the resting Ca2+ level. Whether these basic aspects of presynaptic calcium homeostasis show any consistent trend along the axonal path, and how they are controlled by local network activity, remains poorly understood. Here, we take advantage of the recently advanced FLIM-based method to monitor presynaptic Ca2+ with nanomolar sensitivity. We find that, in cortical pyramidal neurons, action potential-evoked calcium entry (range 10-300 nM), but not the resting Ca2+ level (range 10-100 nM), tends to increase with higher order of axonal branches. Blocking astroglial glutamate uptake reduces evoked Ca2+ entry but has little effect on resting Ca2+ whereas both appear boosted by the constitutive activation of group 1/2 metabotropic glutamate receptors. We find no consistent effect of transient somatic depolarization or hyperpolarization on presynaptic Ca2+ entry or its basal level. The results unveil some key aspects of presynaptic machinery in cortical circuits, shedding light on basic principles of synaptic connectivity in the brain.

Slow-decaying presynaptic calcium dynamics gate long-lasting asynchronous release at the hippocampal mossy fiber to CA3 pyramidal cell synapse.

Action potentials trigger two modes of neurotransmitter release, with a fast synchronous component and a temporally delayed asynchronous release. Asynchronous release contributes to information transfer at synapses, including at the hippocampal mossy fiber (MF) to CA3 pyramidal cell synapse where it controls the timing of postsynaptic CA3 pyramidal neuron firing. Here, we identified and characterized the main determinants of asynchronous release at the MF-CA3 synapse. We found that asynchronous release at MF-CA3 synapses can last on the order of seconds following repetitive MF stimulation. Elevating the stimulation frequency or the external Ca2+ concentration increased the rate of asynchronous release, thus, arguing that presynaptic Ca2+ dynamics is the major determinant of asynchronous release rate. Direct MF bouton Ca2+ imaging revealed slow Ca2+ decay kinetics of action potential (AP) burst-evoked Ca2+ transients. Finally, we observed that asynchronous release was preferentially mediated by Ca2+ influx through P/Q-type voltage-gated Ca2+ channels, while the contribution of N-type VGCCs was limited. Overall, our results uncover the determinants of long-lasting asynchronous release from MF terminals and suggest that asynchronous release could influence CA3 pyramidal cell firing up to seconds following termination of granule cell bursting.

Low-rate firing limit for neurons with axon, soma and dendrites driven by spatially distributed stochastic synapses.

Analytical forms for neuronal firing rates are important theoretical tools for the analysis of network states. Since the 1960s, the majority of approaches have treated neurons as being electrically compact and therefore isopotential. These approaches have yielded considerable insight into how single-cell properties affect network activity; however, many neuronal classes, such as cortical pyramidal cells, are electrically extended objects. Calculation of the complex flow of electrical activity driven by stochastic spatio-temporal synaptic input streams in these structures has presented a significant analytical challenge. Here we demonstrate that an extension of the level-crossing method of Rice, previously used for compact cells, provides a general framework for approximating the firing rate of neurons with spatial structure. Even for simple models, the analytical approximations derived demonstrate a surprising richness including: independence of the firing rate to the electrotonic length for certain models, but with a form distinct to the point-like leaky integrate-and-fire model; a non-monotonic dependence of the firing rate on the number of dendrites receiving synaptic drive; a significant effect of the axonal and somatic load on the firing rate; and the role that the trigger position on the axon for spike initiation has on firing properties. The approach necessitates only calculating the mean and variances of the non-thresholded voltage and its rate of change in neuronal structures subject to spatio-temporal synaptic fluctuations. The combination of simplicity and generality promises a framework that can be built upon to incorporate increasing levels of biophysical detail and extend beyond the low-rate firing limit treated in this paper.

Synaptotagmin 1 oligomers clamp and regulate different modes of neurotransmitter release.

Synaptotagmin 1 (Syt1) synchronizes neurotransmitter release to action potentials (APs) acting as the fast Ca2+ release sensor and as the inhibitor (clamp) of spontaneous and delayed asynchronous release. While the Syt1 Ca2+ activation mechanism has been well-characterized, how Syt1 clamps transmitter release remains enigmatic. Here we show that C2B domain-dependent oligomerization provides the molecular basis for the Syt1 clamping function. This follows from the investigation of a designed mutation (F349A), which selectively destabilizes Syt1 oligomerization. Using a combination of fluorescence imaging and electrophysiology in neocortical synapses, we show that Syt1F349A is more efficient than wild-type Syt1 (Syt1WT) in triggering synchronous transmitter release but fails to clamp spontaneous and synaptotagmin 7 (Syt7)-mediated asynchronous release components both in rescue (Syt1-/- knockout background) and dominant-interference (Syt1+/+ background) conditions. Thus, we conclude that Ca2+-sensitive Syt1 oligomers, acting as an exocytosis clamp, are critical for maintaining the balance among the different modes of neurotransmitter release.

Exact solutions to cable equations in branching neurons with tapering dendrites.

Neurons are biological cells with uniquely complex dendritic morphologies that are not present in other cell types. Electrical signals in a neuron with branching dendrites can be studied by cable theory which provides a general mathematical modelling framework of spatio-temporal voltage dynamics. Typically such models need to be solved numerically unless the cell membrane is modelled either by passive or quasi-active dynamics, in which cases analytical solutions can be reduced to calculation of the Green's function describing the fundamental input-output relationship in a given morphology. Such analytically tractable models often assume individual dendritic segments to be cylinders. However, it is known that dendritic segments in many types of neurons taper, i.e. their radii decline from proximal to distal ends. Here we consider a generalised form of cable theory which takes into account both branching and tapering structures of dendritic trees. We demonstrate that analytical solutions can be found in compact algebraic forms in an arbitrary branching neuron with a class of tapering dendrites studied earlier in the context of single neuronal cables by Poznanski (Bull. Math. Biol. 53(3):457-467, 1991). We apply this extended framework to a number of simplified neuronal models and contrast their output dynamics in the presence of tapering versus cylindrical segments.

Action potential counting at giant mossy fiber terminals gates information transfer in the hippocampus.

Neuronal communication relies on action potential discharge, with the frequency and the temporal precision of action potentials encoding information. Hippocampal mossy fibers have long been recognized as conditional detonators owing to prominent short-term facilitation of glutamate release displayed during granule cell burst firing. However, the spiking patterns required to trigger action potential firing in CA3 pyramidal neurons remain poorly understood. Here, we show that glutamate release from mossy fiber terminals triggers action potential firing of the target CA3 pyramidal neurons independently of the average granule cell burst frequency, a phenomenon we term action potential counting. We find that action potential counting in mossy fibers gates glutamate release over a broad physiological range of frequencies and action potential numbers. Using rapid Ca2+ imaging we also show that the magnitude of evoked Ca2+ influx stays constant during action potential trains and that accumulated residual Ca2+ is gradually extruded on a time scale of several hundred milliseconds. Using experimentally constrained 3D model of presynaptic Ca2+ influx, buffering, and diffusion, and a Monte Carlo model of Ca2+-activated vesicle fusion, we argue that action potential counting at mossy fiber boutons can be explained by a unique interplay between Ca2+ dynamics and buffering at release sites. This is largely determined by the differential contribution of major endogenous Ca2+ buffers calbindin-D28K and calmodulin and by the loose coupling between presynaptic voltage-gated Ca2+ channels and release sensors and the relatively slow Ca2+ extrusion rate. Taken together, our results identify a previously unexplored information-coding mechanism in the brain.

Response functions for electrically coupled neuronal network: a method of local point matching and its applications.

Neuronal networks connected by electrical synapses, also referred to as gap junctions, are present throughout the entire central nervous system. Many instances of gap-junctional coupling are formed between dendritic arbours of individual cells, and these dendro-dendritic gap junctions are known to play an important role in mediating various brain rhythms in both normal and pathological states. The dynamics of such neuronal networks modelled by passive or quasi-active (resonant) membranes can be described by the Green's function which provides the fundamental input-output relationships of the entire network. One of the methods for calculating this response function is the so-called 'sum-over-trips' framework which enables the construction of the Green's function for an arbitrary network as a convergent infinite series solution. Here we propose an alternative and computationally efficient approach for constructing the Green's functions on dendro-dendritic gap junction-coupled neuronal networks which avoids any infinite terms in the solutions. Instead, the Green's function is constructed from the solution of a system of linear algebraic equations. We apply this new method to a number of systems including a simple single cell model and two-cell neuronal networks. We also demonstrate that the application of this novel approach allows one to reduce a model with complex dendritic formations to an equivalent model with a much simpler morphological structure.

Calmodulin as a major calcium buffer shaping vesicular release and short-term synaptic plasticity: facilitation through buffer dislocation.

Action potential-dependent release of synaptic vesicles and short-term synaptic plasticity are dynamically regulated by the endogenous Ca(2+) buffers that shape [Ca(2+)] profiles within a presynaptic bouton. Calmodulin is one of the most abundant presynaptic proteins and it binds Ca(2+) faster than any other characterized endogenous neuronal Ca(2+) buffer. Direct effects of calmodulin on fast presynaptic Ca(2+) dynamics and vesicular release however have not been studied in detail. Using experimentally constrained three-dimensional diffusion modeling of Ca(2+) influx-exocytosis coupling at small excitatory synapses we show that, at physiologically relevant concentrations, Ca(2+) buffering by calmodulin plays a dominant role in inhibiting vesicular release and in modulating short-term synaptic plasticity. We also propose a novel and potentially powerful mechanism for short-term facilitation based on Ca(2+)-dependent dynamic dislocation of calmodulin molecules from the plasma membrane within the active zone.

Differential triggering of spontaneous glutamate release by P/Q-, N- and R-type Ca2+ channels.

The role of voltage-gated Ca2+ channels (VGCCs) in spontaneous miniature neurotransmitter release is incompletely understood. We found that stochastic opening of P/Q-, N- and R-type VGCCs accounts for ∼50% of all spontaneous glutamate release at rat cultured hippocampal synapses, and that R-type channels have a far greater role in spontaneous than in action potential-evoked exocytosis. VGCC-dependent miniature neurotransmitter release (minis) showed similar sensitivity to presynaptic Ca2+ chelation as evoked release, arguing for direct triggering of spontaneous release by transient spatially localized Ca(2+) domains. Experimentally constrained three-dimensional diffusion modeling of Ca2+ influx-exocytosis coupling was consistent with clustered distribution of VGCCs in the active zone of small hippocampal synapses and revealed that spontaneous VGCCs openings can account for the experimentally observed VGCC-dependent minis, although single channel openings triggered release with low probability. Uncorrelated stochastic VGCC opening is therefore a major trigger for spontaneous glutamate release, with differential roles for distinct channel subtypes.

Editorial for special issue on neurodynamics.

"Neurodynamics" is an interdisciplinary area of mathematics where dynamical systems theory (deterministic and stochastic) is the primary tool for elucidating the fundamental mechanisms responsible for the behaviour of neural systems (whether biological or synthetic). A meeting on this topic was held at the International Centre for Mathematical Sciences in Edinburgh from March 5-7 in 2012. In this special issue, we have invited seven of the main contributors to this event to expand on their presentations and highlight the use of mathematics in understanding the dynamics of neural systems.

Gap junctions, dendrites and resonances: a recipe for tuning network dynamics.

Gap junctions, also referred to as electrical synapses, are expressed along the entire central nervous system and are important in mediating various brain rhythms in both normal and pathological states. These connections can form between the dendritic trees of individual cells. Many dendrites express membrane channels that confer on them a form of sub-threshold resonant dynamics. To obtain insight into the modulatory role of gap junctions in tuning networks of resonant dendritic trees, we generalise the "sum-over-trips" formalism for calculating the response function of a single branching dendrite to a gap junctionally coupled network. Each cell in the network is modelled by a soma connected to an arbitrary structure of dendrites with resonant membrane. The network is treated as a single extended tree structure with dendro-dendritic gap junction coupling. We present the generalised "sum-over-trips" rules for constructing the network response function in terms of a set of coefficients defined at special branching, somatic and gap-junctional nodes. Applying this framework to a two-cell network, we construct compact closed form solutions for the network response function in the Laplace (frequency) domain and study how a preferred frequency in each soma depends on the location and strength of the gap junction.

Computational convergence of the path integral for real dendritic morphologies.

Neurons are characterised by a morphological structure unique amongst biological cells, the core of which is the dendritic tree. The vast number of dendritic geometries, combined with heterogeneous properties of the cell membrane, continue to challenge scientists in predicting neuronal input-output relationships, even in the case of sub-threshold dendritic currents. The Green's function obtained for a given dendritic geometry provides this functional relationship for passive or quasi-active dendrites and can be constructed by a sum-over-trips approach based on a path integral formalism. In this paper, we introduce a number of efficient algorithms for realisation of the sum-over-trips framework and investigate the convergence of these algorithms on different dendritic geometries. We demonstrate that the convergence of the trip sampling methods strongly depends on dendritic morphology as well as the biophysical properties of the cell membrane. For real morphologies, the number of trips to guarantee a small convergence error might become very large and strongly affect computational efficiency. As an alternative, we introduce a highly-efficient matrix method which can be applied to arbitrary branching structures.

Mixed mode oscillations in mouse spinal motoneurons arise from a low excitability state.

We explain the mechanism that elicits the mixed mode oscillations (MMOs) and the subprimary firing range that we recently discovered in mouse spinal motoneurons. In this firing regime, high-frequency subthreshold oscillations appear a few millivolts below the spike voltage threshold and precede the firing of a full blown spike. By combining intracellular recordings in vivo (including dynamic clamp experiments) in mouse spinal motoneurons and modeling, we show that the subthreshold oscillations are due to the spike currents and that MMOs appear each time the membrane is in a low excitability state. Slow kinetic processes largely contribute to this low excitability. The clockwise hysteresis in the I-F relationship, frequently observed in mouse motoneurons, is mainly due to a substantial slow inactivation of the sodium current. As a consequence, less sodium current is available for spiking. This explains why a large subprimary range with numerous oscillations is present in motoneurons displaying a clockwise hysteresis. In motoneurons whose I-F curve exhibits a counterclockwise hysteresis, it is likely that the slow inactivation operates on a shorter time scale and is substantially reduced by the de-inactivating effect of the afterhyperpolarization (AHP) current, thus resulting in a more excitable state. This accounts for the short subprimary firing range with only a few MMOs seen in these motoneurons. Our study reveals a new role for the AHP current that sets the membrane excitability level by counteracting the slow inactivation of the sodium current and allows or precludes the appearance of MMOs.

Intercellular calcium waves in the fire-diffuse-fire framework: Green's function for gap-junctional coupling.

Calcium is a crucial component in a plethora of cellular processes involved in cell birth, life, and death. Intercellular calcium waves that can spread through multiple cells provide one form of cellular communication mechanism between various parts of cell tissues. Here we introduce a simple, yet biophysically realistic model for the propagation of intercellular calcium waves based on the fire-diffuse-fire type model for calcium dynamics. Calcium release sites are considered to be discretely distributed along individual linear cells that are connected by gap junctions and a solution of this model can be found in terms of the Green's function for this system. We develop the "sum-over-trips" formalism that takes into account the boundary conditions at gap junctions providing a generalization of the original sum-over-trips approach for constructing the response function for branched neural dendrites. We obtain the exact solution of the Green's function in the Laplace (frequency) domain for an infinite array of cells and show that this Green's function can be well approximated by its truncated version. This allows us to obtain an analytical traveling wave solution for an intercellular calcium wave and analyze the speed of solitary wave propagation as a function of physiologically important system parameters. Periodic and irregular traveling waves can be also sustained by the proposed model.