THE RESONATE-AND-FIRE NEURON: TIME DEPENDENT AND FREQUENCY SELECTIVE NEURONS IN NEURAL NETWORKS

The means through which the nervous system perceives its environment is one of the most fascinating questions in contemporary science. Our endeavors to comprehend the principles of neural science provide an instance of how biological processes may inspire novel methods in mathematical modeling and engineering. The application ofmathematical models towards understanding neural signals and systems represents a vibrant field of research that has spanned over half a century. During this period, multiple approaches to neuronal modeling have been adopted, and each approach is adept at elucidating a specific aspect of nervous system function. Thus while bio-physical models have strived to comprehend the dynamics of actual physical processes occurring within a nerve cell, the phenomenological approach has conceived models that relate the ionic properties of nerve cells to transitions in neural activity. Further-more, the field of neural networks has endeavored to explore how distributed parallel processing systems may become capable of storing memory. Through this project, we strive to explore how some of the insights gained from biophysical neuronal modeling may be incorporated within the field of neural net-works. We specifically study the capabilities of a simple neural model, the Resonate-and-Fire (RAF) neuron, whose derivation is inspired by biophysical neural modeling. While reflecting further biological plausibility, the RAF neuron is also analytically tractable, and thus may be implemented within neural networks. In the following thesis, we provide a brief overview of the different approaches that have been adopted towards comprehending the properties of nerve cells, along with the framework under which our specific neuron model relates to the field of neuronal modeling. Subsequently, we explore some of the time-dependent neurocomputational capabilities of the RAF neuron, and we utilize the model to classify logic gates, and solve the classic XOR problem. Finally we explore how the resonate-and-fire neuron may be implemented within neural networks, and how such a network could be adapted through the temporal backpropagation algorithm.

Biaxial extensional motion of an inertially driven radially expanding liquid sheet

We consider the inertially driven, time-dependent biaxial extensional motion of inviscid and viscous thinning liquid sheets. We present an analytic solution describing the base flow and examine its linear stability to varicose (symmetric) perturbations within the framework of a long-wave model where transient growth and long-time asymptotic stability are considered. The stability of the system is characterized in terms of the perturbation wavenumber, Weber number, and Reynolds number. We find that the isotropic nature of the base flow yields stability results that are identical for axisymmetric and general two-dimensional perturbations. Transient growth of short-wave perturbations at early to moderate times can have significant and lasting influence on the long-time sheet thickness. For finite Reynolds numbers, a radially expanding sheet is weakly unstable with bounded growth of all perturbations, whereas in the inviscid and Stokes flow limits sheets are unstable to perturbations in the short-wave limit.

Assessing the Validity of Brain Glucose Concentration Measurement Using Microdialysis

Brain functions, such as learning, orchestrating locomotion, memory recall, and processing information, all require glucose as a source of energy. During these functions, the glucose concentration decreases as the glucose is being consumed by brain cells. By measuring this drop in concentration, it is possible to determine which parts of the brain are used during specific functions and consequently, how much energy the brain requires to complete the function. One way to measure in vivo brain glucose levels is with a microdialysis probe. The drawback of this analytical procedure, as with many steadystate fluid flow systems, is that the probe fluid will not reach equilibrium with the brain fluid. Therefore, brain concentration is inferred by taking samples at multiple inlet glucose concentrations and finding a point of convergence. The goal of this thesis is to create a three-dimensional, time-dependent, finite element representation of the brainprobe system in COMSOL 4.2 that describes the diffusion and convection of glucose. Once validated with experimental results, this model can then be used to test parameters that experiments cannot access. When simulations were run using published values for physical constants (i.e. diffusivities, density and viscosity), the resulting glucose model concentrations were within the error of the experimental data. This verifies that the model is an accurate representation of the physical system. In addition to accurately describing the experimental brain-probe system, the model I created is able to show the validity of zero-net-flux for a given experiment. A useful discovery is that the slope of the zero-net-flux line is dependent on perfusate flow rate and diffusion coefficients, but it is independent of brain glucose concentrations. The model was simplified with the realization that the perfusate is at thermal equilibrium with the brain throughout the active region of the probe. This allowed for the assumption that all model parameters are temperature independent. The time to steady-state for the probe is approximately one minute. However, the signal degrades in the exit tubing due to Taylor dispersion, on the order of two minutes for two meters of tubing. Given an analytical instrument requiring a five μL aliquot, the smallest brain process measurable for this system is 13 minutes.