A Simple Mathematical Model Inspired by the Purkinje Cells: From Delayed Travelling Waves to Fractional Diffusion

Synaptic propagation in neuronal networks with finite-support space-dependent coupling

Traveling waves of electrical activity are ubiquitous in biological neuronal networks. Traveling waves in the brain are associated with sensory processing, phase coding, and sleep. The neuron and network parameters that determine traveling waves' evolution are the synaptic space constant, synaptic conductance, membrane time constant, and synaptic decay time constant. We used an abstract neuron model in a one-dimensional network to investigate the propagation characteristics of traveling wave activity. We formulate a set of evolution equations based on the network connectivity parameters. Using a combination of numerical and analytical approaches, we show that these traveling waves are stable to a series of perturbations with biological relevance.

Label-free Optical Imaging of Membrane Potential

Offering high temporal resolution, voltage imaging is an important and essential technique in neuroscience. Among different optical imaging approaches, the label-free approach remains attractive due to its unique value coming from free of exogenous chromophores. The intrinsic voltage-indicating signals arising from membrane deformation, membrane spectral change, phase shift, light scattering, and membrane hydration haven been reported. First demonstrated 70 years ago, label-free optical imaging of membrane potential is still at an early stage and the field is challenged by the relatively small signals generated by the intrinsic optical properties. We review major contrast mechanisms used for label-free voltage imaging and discuss several recent exciting advances that could potentially enable membrane potential imaging in mammalian neurons at high speed and high sensitivity.

Existence and Stability Analysis for Fractional Differential Equations with Mixed Nonlocal Conditions

In this paper, we study the existence and uniqueness of solution for fractional differential equations with mixed fractional derivatives, integrals and multi-point conditions. After that, we also establish different kinds of Ulam stability for the problem at hand. Examples illustrating our results are also presented.

Interior Regularity Estimates for a Degenerate Elliptic Equation with Mixed Boundary Conditions

The Marchaud fractional derivative can be obtained as a Dirichlet-to–Neumann map via an extension problem to the upper half space. In this paper we prove interior Schauder regularity estimates for a degenerate elliptic equation with mixed Dirichlet–Neumann boundary conditions. The degenerate elliptic equation arises from the Bernardis–Reyes–Stinga–Torrea extension of the Dirichlet problem for the Marchaud fractional derivative.