Multi-scale modeling and asymptotic analysis for neuronal synapses and networks (original) (raw)
Related papers
Modélisation multi-échelle et analyse asymptotique pour les synapses et les réseaux neuronaux
2015
In the present PhD thesis, we study neuronal structures at different scales, from synapses to neural networks. Our goal is to develop mathematical models and their analysis, in order to determine how the properties of synapses at the molecular level shape their activity and propagate to the network level. This change of scale can be formulated and analyzed using several tools such as partial differential equations, stochastic processes and numerical simulations. In the first part, we compute the mean time for a Brownian particle to arrive at a narrow opening defined as the small cylinder joining two tangent spheres. The method relies on Möbius conformal transformation applied to the Laplace equation. We also estimate, when the particle starts inside a boundary layer near the hole, the splitting probability to reach the hole before leaving the boundary layer, which is also expressed using a mixed boundary-value Laplace equation. Using these results, we develop model equations and the...
Stochastic lattice model of synaptic membrane protein domains
Physical Review E, 2017
Neurotransmitter receptor molecules, concentrated in synaptic membrane domains along with scaffolds and other kinds of proteins, are crucial for signal transmission across chemical synapses. In common with other membrane protein domains, synaptic domains are characterized by low protein copy numbers and protein crowding, with rapid stochastic turnover of individual molecules. We study here in detail a stochastic lattice model of the receptor-scaffold reaction-diffusion dynamics at synaptic domains that was found previously to capture, at the mean-field level, the self-assembly, stability, and characteristic size of synaptic domains observed in experiments. We show that our stochastic lattice model yields quantitative agreement with mean-field models of nonlinear diffusion in crowded membranes. Through a combination of analytic and numerical solutions of the master equation governing the reaction dynamics at synaptic domains, together with kinetic Monte Carlo simulations, we find substantial discrepancies between mean-field and stochastic models for the reaction dynamics at synaptic domains. Based on the reaction and diffusion properties of synaptic receptors and scaffolds suggested by previous experiments and mean-field calculations, we show that the stochastic reaction-diffusion dynamics of synaptic receptors and scaffolds provide a simple physical mechanism for collective fluctuations in synaptic domains, the molecular turnover observed at synaptic domains, key features of the observed single-molecule trajectories, and spatial heterogeneity in the effective rates at which receptors and scaffolds are recycled at the cell membrane. Our work sheds light on the physical mechanisms and principles linking the collective properties of membrane protein domains to the stochastic dynamics that rule their molecular components.
Physical Review E, 2009
Spike packet propagation is modeled in mesoscopic-scale networks, composed of locally and recurrently coupled neural pools, and embedded in a two-dimensional lattice. Site dynamics is governed by three key parameters-pool connectedness probability, synaptic strength ͑following the steady-state distribution of some realizations of spike-timing-dependent plasticity learning rule͒, and the neuron refractoriness. Formation of spatiotemporal patterns in our model, synfire chains, exhibits critical behavior, with the emerging percolation phase transition controlled by the probability for nonzero synaptic strength value. Applying the finite-size scaling method, we infer the critical probability dependence on synaptic strength and refractoriness and determine the effects of connection topology on the pertaining percolation clusters fractal dimensions. We find that the directed percolation and the pair contact process with diffusion constitute the relevant universality classes of phase transitions observed in a class of mesoscopic-scale network models, which may be related to recently reported data on in vitro cultures.
Search Time for a Small Ribbon and Application to Vesicular Release at Neuronal Synapses
Multiscale Modeling & Simulation, 2015
The arrival of a Brownian particle at a narrow cusp located underneath a ball is a model of vesicular release at neuronal synapses, triggered by calcium ions. The asymptotic computation of the arrival time presents several difficulties that can be overcome using conformal mappings and asymptotic analysis of the model equations. Using a regular expansion of the solution of the Laplace equation in the mapped domain, we compute the solution involving both small and large spatial scales. We derive novel asymptotic formulas for Brownian escape through cusps in both two and three dimensions. The range of validity of the asymptotic formulas is challenged by stochastic simulations. Finally, we apply the analysis to estimate the vesicular release probability at presynaptic terminals and, in particular, we suggest that vesicular organization imposes a severe constraint on calcium channel localization: diffusing calcium ions can trigger vesicular release only in a specific range of positions that we provide.
Growth-driven percolations: the dynamics of connectivity in neuronal systems
The European Physical Journal B, 2005
The quintessential property of neuronal systems is their intensive patterns of selective synaptic connections. The current work describes a physics-based approach to neuronal shape modeling and synthesis and its consideration for the simulation of neuronal development and the formation of neuronal communities. Starting from images of real neurons, geometrical measurements are obtained and used to construct probabilistic models which can be subsequently sampled in order to produce morphologically realistic neuronal cells. Such cells are progressively grown while monitoring their connections along time, which are analysed in terms of percolation concepts. However, unlike traditional percolation, the critical point is verified along the growth stages, not the density of cells, which remains constant throughout the neuronal growth dynamics. It is shown, through simulations, that growing beta cells tend to reach percolation sooner than the alpha counterparts with the same diameter. Also, the percolation becomes more abrupt for higher densities of cells, being markedly sharper for the beta cells. In the addition to the importance of the reported concepts and methods to computational neuroscience, the possibility of reaching percolation through morphological growth of a fixed number of objects represents in itself a novel paradigm of great theoretical and practical interest for the areas of statistical physics and critical phenomena.
Stochastic and reduced biophysical models of synaptic transmission
Journal of biological physics, 2000
Stochastic and reduced biophysical models of synaptictransmission are formulated and evaluated. Thesynaptic transmission involves presynapticfacilitation of neurotransmitter release, depletionand recovery of the presynaptic pool of readilyreleasable vesicles containing neurotransmittermolecules and saturation of postsynaptic receptors ofboth fast non-NMDA and slow NMDA types. The models areshown to display the principal dynamicalcharacteristics experimentally observed of synaptictransmission. The two main types of neural coding,i.e. rate and temporal coding, can be distinguished bymeans of different dynamical properties of synaptictransmission determined by initial neurotransmitterrelease probability and presynaptic firing rate. Fromthe temporal evolution of the postsynaptic membranepotential response to a train of presynaptic actionpotentials at a sustained firing rate, in particularthe steady-state amplitude and steady-state averagelevel of postsynaptic membrane potentials aredete...
ArXiv, 2020
There is a strong nexus between the network size and the computational resources available, which may impede a neuroscience study. In the meantime, rescaling the network while maintaining its behavior is not a trivial mission. Additionally, modeling patterns of connections under topographic organization presents an extra challenge: to solve the network boundaries or mingled with an unwished behavior. This behavior, for example, could be an inset oscillation due to the torus solution; or a blend with/of unbalanced neurons due to a lack (or overdose) of connections. We detail the network rescaling method able to sustain behavior statistical utilized in Romaro et al. (2018) and present a boundary solution method based on the previous statistics recoup idea.
Cooperative stochastic binding and unbinding explain synaptic size dynamics and statistics
PLoS Computational Biology, 2017
Synapses are dynamic molecular assemblies whose sizes fluctuate significantly over timescales of hours and days. In the current study, we examined the possibility that the spontaneous microscopic dynamics exhibited by synaptic molecules can explain the macroscopic size fluctuations of individual synapses and the statistical properties of synaptic populations. We present a mesoscopic model, which ties the two levels. Its basic premise is that synaptic size fluctuations reflect cooperative assimilation and removal of molecules at a patch of postsynaptic membrane. The introduction of cooperativity to both assimilation and removal in a stochastic biophysical model of these processes, gives rise to features qualitatively similar to those measured experimentally: nanoclusters of synaptic scaffolds, fluctuations in synaptic sizes, skewed, stable size distributions and their scaling in response to perturbations. Our model thus points to the potentially fundamental role of cooperativity in dictating synaptic remodeling dynamics and offers a conceptual understanding of these dynamics in terms of central microscopic features and processes.
Boundary conditions and phase transitions in neural networks. Simulation results
Neural Networks, 2008
This paper gives new simulation results on the asymptotic behaviour of theoretical neural networks on Z and Z 2 following an extended Hopfield law. It specifically focuses on the influence of fixed boundary conditions on such networks. First, we will generalise the theoretical results already obtained for attractive networks in one dimension to more complicated neural networks. Then, we will focus on twodimensional neural networks. Theoretical results have already been found for the nearest neighbours Ising model in 2D with translation-invariant local isotropic interactions. We will detail what happens for this kind of interaction in neural networks and we will also focus on more complicated interactions, i.e., interactions that are not local, neither isotropic, nor translation-invariant. For all these kinds of interactions, we will show that fixed boundary conditions have significant impacts on the asymptotic behaviour of such networks. These impacts result in the emergence of phase transitions whose geometric shape will be numerically characterised.