Mathematical Biology Seminar - Fall 2024

For questions about the seminar schedule, please contact James MacLaurin.

September 11

Bryan Currie, NJIT

Maximum Increase in Widths and Algebra with Quintet Information

Mathematical phylogenists are interested in improving models for evolution, and our current models tend to spit out trees that seem more balanced than trees we infer from data. As tree balance is ambiguous, there are several tree balance indices which capture different notions of balance. One less-understood tree balance index is maximum increase in widths when limited to binary trees. We partially fill in a gap in the literature by describing the structure and size of maximal trees for leaf numbers that are not powers of two.
As for the quintets, it is well-known that data of unrooted quartets among taxa can determine the unrooted tree topology under the multispecies coalescent model of evolution. However, these data cannot determine the root of the tree, and so biologists have resorted to including an outgroupspecies, which is significantly less related to the others in the study that split off from a common ancestor of the entire relevant clade. While a reasonable idea, this leads to problems if the outgroup chosen is too closely or too distantly related. Unrooted quintet data had already been shown to identify the root for a binary tree, but we used an algebraic approach to reprove this and also show that quintet data can also detect network features invisible to quartet data, such as rooting in some cases and 2-cycles in some cases.

September 18

James MacLaurin, NJIT

An Introduction to the Coupling Method (for deriving Fokker-Planck population density PDEs from particle systems)

Mathematical phylogenists are interested in improving models for evolution, and our current models tend to spit out trees that seem more balanced than trees we infer from data. As tree balance is ambiguous, there are several tree balance indices which capture different notions of balance. One less-understood tree balance index is maximum increase in widths when limited to binary trees. We partially fill in a gap in the literature by describing the structure and size of maximal trees for leaf numbers that are not powers of two.
As for the quintets, it is well-known that data of unrooted quartets among taxa can determine the unrooted tree topology under the multispecies coalescent model of evolution. However, these data cannot determine the root of the tree, and so biologists have resorted to including an outgroupspecies, which is significantly less related to the others in the study that split off from a common ancestor of the entire relevant clade. While a reasonable idea, this leads to problems if the outgroup chosen is too closely or too distantly related. Unrooted quintet data had already been shown to identify the root for a binary tree, but we used an algebraic approach to reprove this and also show that quintet data can also detect network features invisible to quartet data, such as rooting in some cases and 2-cycles in some cases.

September 25

Shahriar Afkhami, NJIT

Numerical simulations and physics-informed neural networks for magnetic drug delivery

Magnetic drug delivery is a promising method for non-invasive, targeted treatment for certain diseases, including cancer, particularly for targeting specific, hard-to-reach sites in the body, resulting in reducing side effects by lowering dosages. This is achieved by injecting superparamagnetic iron oxide nanoparticles containing therapeutic drugs into the bloodstream and using an external magnet to guide these nanoparticles to the tumor site. I will present a computational model consisting of a stochastic ordinary differential equation model to simulate nanoparticle trajectories in blood flow and to determine the optimal conditions for magnetic drug delivery. I will also present a physics-informed neural network approach as an efficient machine learning method for solving the governing initial value differential equation.

October 9

Andrew White, NJIT

Role of calcium buffers in synaptic neurotransmitter release and its short-term modulation

Neurotransmitter release is caused by a series of biological steps. It starts with an action potential traveling down the axon and inducing the voltage gated calcium ion channels to open and induces an influx of calcium ions into the pre-synaptic terminal. Following this, calcium ions bind to sensor proteins signaling the vesicles to fuse with the inner pre-synaptic membrane which release their contents into the synaptic cleft. The efficiency of neurotransmitter release is not constant, but changes from pulse to pulse and it can either increase (short term facilitation or STF) or decrease (short term depression or STD). Collectively, these behaviors are referred to as short-term synaptic plasticity (STSP).

Our aim is to understand how the properties of intracellular calcium-binding molecules called calcium buffers affect STSP. To address this question, we solve the corresponding system of reaction-diffusion PDEs numerically (via a Crank-Nicholson scheme) and explore interesting STSP behaviors elicited in different parameter regimes. In our research we are focusing on calcium buffers that have two calcium ion binding sites per buffer molecule. To find optimal buffering properties associated with each specific STSP behavior that we observe, we employ a differential evolution optimization algorithm. Finally, we discuss possible extensions of this work to the study of other calcium dependent mechanisms such as intracellular calcium waves caused by calcium-induced calcium release, known to be strongly dependent on buffering properties.

October 16

Horacio Rotstein, NJIT

Mathematical Degeneracy Meets Biological Degeneracy: A discussion on the Implications for Parameters Estimation Identifiability and Dynamical Systems Reconstruction

Parameter estimation from observable or experimental data is a crucial stage in any modeling study. The ability of models to make predictions, provide mechanistic explanations, and be useful for decision making all depends on the accuracy and reliability of the parameter estimation process. Several difficulties conspire against our ability to successfully achieve this goal. These difficulties are data-related (lack of access to all state variables, inconsistent gaps across trials), computational (algorithmic nature), statistical (data is noisy and therefore one can at best expect to estimate distributions of parameter values around a ``true" mean), and structural (degeneracy, mathematical nature). Identifiability refers to one’s ability to uniquely estimate the model parameters from the available data. Unidentifiability in dynamic models, the opposite of identifiability, is associated with the notion of degeneracy where multiple parameter sets produce the same pattern, therefore the inverse function of determining the model parameters from the data is not well defined. Degeneracy is not only a mathematical property of models, but it has also been reported in biological experiments of neuronal oscillations and is likely to be identified in other biological systems.

The goal of this talk is to discussion the various forms of degeneracy and unidentifiability (the other side of the coin), their connection to other related phenomena such as redundancy, and the implications for parameters estimation using dynamic models, the development of parameter estimation algorithms and the development of algorithmic tools (e.g., machine learning) for data-driven discovery of nonlinear dynamics governing the generation of these data.

October 23

Joshua McGinnis, University of Pennsylvania (UPenn)

Homogenization of a Spatially Extended, Stochastic Ion Channel Model

Simulations of stochastic neuron potential models, which describe the voltage potential along the length of a neuron’s axon and incorporate ion channel noise as Gaussian fluctuations, have shown that channel noise can induce complex phenomena such as jitters and splitting of action potentials [1] and place constraints on the miniaturization of axons [2]. To develop a robust analytic framework for understanding stochastic effects of channel noise on action potential propagation in a neuron, we need to begin by investigating how many independent, spatially distributed ion channels can collectively yield deterministic behavior. We start with an electrophysiological derivation of a simple discrete model and contrast this with a common, yet less physically accurate approach where the law of large numbers and the central limit theorem are more easily applied. Our model couples a spatially discretized diffusive PDE for the voltage with continuous-time Markov processes that govern the behavior of the ion channels. We will then outline an argument using homogenization theory to estimate the rate of strong convergence to the typical deterministic PDE as the spacing between ion channels approaches zero. Finally, we present a numerical technique for simulating our model and discuss the challenges involved in increasing computational efficiency of simulations.

[1] Faisal AA, Laughlin SB. Stochastic simulations on the reliability of action potential propagation in thin axons. PLoS Comput Biol. 2007 May;3(5):e79. doi: 10.1371/journal.pcbi.0030079. PMID: 17480115; PMCID: PMC1864994.
[2] Faisal AA, White JA, Laughlin SB. Ion-channel noise places limits on the miniaturization of the brain's wiring. Curr Biol. 2005 Jun 21;15(12):1143-9. doi: 10.1016/j.cub.2005.05.056. PMID: 15964281.

October 30

Mark Van Den Bosch, Leiden University

Multidemensional Stability of Planar Travelling Waves for Stochastically Perturbed Reaction-Diffusion Systems

We consider reaction-diffusion systems with multiplicative noise on a spatial domain of dimension two or higher. The noise process is white in time, coloured in space, and invariant under translations. Inspired by previous works on the real line, we establish the multidimensional stability of planar waves on a cylindrical domain on time scales that are exponentially long with respect to the noise strength. This is achieved by means of a stochastic phase tracking mechanism that can be maintained over such long time scales. The corresponding mild formulation of our problem features stochastic integrals with respect to anticipating integrands, which hence cannot be understood within the well-established setting of Itˆo-integrals. To circumvent this problem, we exploit and extend recently developed theory concerning forward integrals.

November 6

Rik Westdorp, Leiden University

Stochastic Soliton Dynamics in the Korteweg-De Vries Equation with Multiplicative Noise

In recent years, stochastic traveling waves have become a major area of interest in the field of stochastic PDEs. Various approaches have been to introduced to study the effects of noise on traveling waves, mainly in the setting of Reaction-Diffusion equations. Of particular interest is the notion of a stochastic wave position and its dynamics. This talk focusses on solitary waves in the Korteweg-de Vries equation. Due to a scaling symmetry, this dispersive PDE supports a solitary wave family of various amplitudes and velocities. We introduce stochastic processes that track the amplitude and position of solitons under the influence of multiplicative noise over long time-scales. Our method is based on a rescaled frame and stability properties of the solitary waves. We formulate expansions for the stochastic soliton amplitude and position, and compare their leading-order dynamics with numerical simulations.

This is joint work with Prof. H.J. Hupkes

November 13

Rainer Engelken, Columbia University (Center for Theoretical Neuroscience)

Sparse Chaos in Cortical Circuits

Nerve impulses, the currency of information flow in the brain, are generated by an instability of the neuronal membrane potential dynamics. Neuronal circuits exhibit collective chaos that appears essential for learning, memory, sensory processing, and motor control. However, the factors controlling the nature and intensity of collective chaos in neuronal circuits are not well understood. Here we use computational ergodic theory to demonstrate that basic features of nerve impulse generation profoundly affect collective chaos in neuronal circuits. Numerically exact calculations of Lyapunov-spectra, Kolmogorov-Sinai-entropy, and upper and lower bounds on attractor dimension show that changes in nerve impulse generation in individual neurons only moderately modify the rate of information encoding but qualitatively transform phase space structure, reducing the number of unstable manifolds, Kolgomorov-Sinai-entropy, and attractor dimension by orders of magnitude. Beyond a critical point, marked by a localization transition of the leading covariant Lyapunov vector which coincides with the breakdown of the diffusion approximation, the networks exhibit sparse chaos: extended periods of near stable dynamics interrupted by short bursts of intense chaos. Analysis of large networks with a more realistic structure indicates the generality of these findings. In cortical circuits, biophysics appears tuned to this regime of sparse chaos. Our results demonstrate a tight link between fundamental features of single-neuron biophysics and the collective dynamics of cortical circuits and suggest that the machinery of nerve impulse generation is tailored to enhance circuit controllability and information flow.

November 20

Allison Edgar, NJIT

Ctenophore life history in the lab: can we teach an old animal new tricks?

Animals have evolved diverse strategies to reproduce, develop, grow, and survive, known as “life histories”. Our lab uses an emerging animal model whose unique suite of traits makes them uniquely suited to laboratory investigation of life history traits, but which have been historically under-studied, the marine invertebrates known as “comb jellies” or ctenophores. Ctenophores are the likely sister group of all other living animals. Their accessible embryos, capacity for whole-body regeneration, and life-long optical transparency make them well suited to laboratory life, while several recently published papers suggest that their life history may be particularly informative for understanding the origins of animal life cycles. My new lab at NJIT is interested in understanding 1) how environmental inputs affect ctenophores’ reproduction and regeneration, 2) how body shape changes occur throughout the lifecycle and whether these morphological changes are correlated to other kinds of changes, and 3) identifying the cellular and gene regulatory basis of these changes. I will focus my talk on aspects of ctenophore biology I think are particularly likely to spark active collaboration with mathematicians, including pilot data from projects begun in my new lab at NJIT.

December 4

Farzan Nadim, NJIT

Parsing Cerebellar Signals to the Substantia Nigra Dopamine System

Abstract Forthcoming