Stochastic and Mean-Field Analysis in Neuroscience

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James MacLaurin

"Coarse-Graining of the Primary Visual Cortex"
Experimental evidence indicates that the dynamic response of the primary visual cortex to stimulii continues over relatively slow timescales of O(100ms). Typically, the initial 'bump' of activity, that is centered at a particular orientation, slowly softens and dissipates after the stimulus is removed. To explain this phenomenon, we develop a realistic microscopic neural field model, and perform a coarse-graining to obtain a macroscopic neural field equation. The microscopic model contains slow synaptic dynamics, and stochasticity resulting from synaptic transmission failure. At the macroscopic level, after a coarse-graining procedure, this stochasticity results in a neural field equation with an integral convolution (as is standard in neural field equations) but also a spatial Laplacian, and with the diffusion coefficient proportional to the population activity. Our macroscopic equation can be thought of as a spatially-extended population density equation: it combines the strengths of neural-field equations and population density equations into a single formalism. It allows a deeper understanding of how changes in the average synchronization of neurons affects the macroscopic dynamics. It partly parallels the efforts of Coombes et al in recent years to derive such 'next-generation neural field equations'. We next perform a Large Deviations analysis to study the typical stochastic fluctuations about the limiting equation. This allows us to determine the most likely abnormal behavior that could be induced in the system by finite size effects. This work is based on a preprint (joint with Bart Krekelberg) entitled 'Coarse-Graining of Neural Networks with Stochastic Dynamic Connections.'

Youngmin Park

Brandies U
"Dynamics of Vesicles Driven into Closed Constrictions by Molecular Motors"
We study the dynamics of a model of membrane vesicle transport into dendritic spines, which are bulbous intracellular compartments in neurons driven by molecular motors. We explore the effects of noise on the reduced lubrication model proposed in (Fai et al, Active elastohydrodynamics of vesicles in narrow, blind constrictions. Phys. Rev. Fluids, 2 (2017), 113601). The Fokker-Planck approximation fails to capture mean first passage times of velocity switching (tug-of-war effect), and the agent-based model is computationally expensive. For relatively efficient computations, we turn to the master equation and find that it requires an additional calculation to account for non-equilibrium dynamics in the underlying myosin motor population. We discuss remaining questions and future directions in this ongoing work.

Victor Matveev

"Mass-Action vs Stochastic Modeling of First Passage Time to Ca2+-Triggered Vesicle Release"
Like most physiological cell mechanisms, synaptic neurotransmitter vesicle release (exocytosis) is characterized by a high degree of variability in all steps of the process, from Ca2+ channel gating to the final triggering of membrane fusion by the SNARE machinery. The associated fluctuations can be quite large since only a small number of Ca2+ ions enter the cell through a single channel during an action potential, and further increased by the stochasticity in the Ca2+ binding to Ca2+ buffers and sensors. This leads to a widely-held assumption that solving mass-action reaction-diffusion equations for buffered Ca2+ diffusion does not provide sufficient insight into the underlying Ca2+-dependent cell processes. However, several comparative studies showed a surprisingly close agreement between deterministic and trial-averaged stochastic simulations of Ca2+ diffusion, buffering and binding, as long as Ca2+ channel gating is not Ca2+ dependent. We present further comparison of stochastic and mass-action simulations, focusing on Ca2+ dynamics downstream of Ca2+ channel gating and considering spatially-resolved reaction-diffusion modeling in 3D. Namely, we compare the distributions of first-passage-times (FPT) to full binding of the model Ca2+ sensor for vesicle fusion obtained using stochastic and deterministic approaches. We note that in the deterministic formulation, FPT density is equivalent to the time-dependent rate of the final irreversible transition to the fusion-ready sensor state. We show that the discrepancy between deterministic and stochastic approaches in simulating the FPT density can be surprisingly small even when only as few as 40 ions enter the cell per single channel-vesicle complex, despite the fluctuations caused by the Ca2+ binding and unbinding. Further, we demonstrate this close agreement between stochastic and deterministic FPT computation using a highly simplified two-compartment model, whereby the FPT density can be computed exactly using either of the two approaches. The reason for the close agreement between the two methods is that in the absence of Ca2+-induced Ca2+-release, the non-linearities in the exocytosis process involve only bi-molecular reactions. Therefore, the discrepancy between the two approaches is primarily determined by the size of correlations between reactant molecule number fluctuations rather than the fluctuation amplitudes. The small size of reactant correlations is in turn determined by the relationship between the rate of diffusion relative the rate of Ca2+ buffering and binding, as suggested in prior studies. In most common parameter regimes, FPT density is not very sensitive to the fluctuations in the rates of Markovian transitions between distinct sensor states arising from the Ca2+ fluctuations, which in turn leads to small discrepancies between the two approaches. This work is supported by NSF grant DMS-1517085.

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