Mathematical Biology Seminar - Spring 2017

Canards are special solutions of slow/fast systems that spend long times near repelling sets. They are ubiquitous in applications and have been used to explain the firing patterns of electrically excitable cells in neuroscience, the sudden change in amplitude and period of oscillatory behaviour in chemical reactions, and the anomalous delays in response to exogenous pulses of inositol triphosphate in calcium signalling. In this talk, we survey some of the exciting new developments in canard theory and its applications to bursting in nerve and endocrine cells. In particular, we devise analytic criteria for the detection and classification of a new class of singularities for differential equations, which act as organizing centers for complex oscillatory behaviour. We demonstrate our theory in a model for intracellular calcium dynamics, where we discover a new class of bursting rhythms and explain the mechanisms that underlie them.

Methods of data assimilation, such as the ensemble Kalman filter, have recently been applied to tracking neuronal network structure parameters. This is a particularly challenging domain for parameter estimation due to the strongly non-linear models combined with large observation noise, a high-dimensional parameter space, and model uncertainty. We discuss recent practical improvements in adaptive filtering, and alternative approaches when model error is large or parts of the model are unknown. In the extreme case that no model is available, we propose a method using delay coordinate reconstruction that merges Kalman filtering with Takens' work on nonlinear data analysis.

Perceptual bistability arises from ambiguous stimuli that our brain cannot recognise univocally. Under a steady sensory input, subjects usually report alternation of percepts resulting in a sequence of single-percept time intervals that typically follow a Gamma distribution. Different models have been proposed to account for this statistics of dominance times, most of them containing few variables that represent key features of the main involved areas and, eventually, some slow adaptation processes. Two-attractor models have been extensively studies but, recently, heteroclinic networks have successfully been applied to model this phenomenon, see [Ashwin and Lavric, Physica D (2010)]. The mechanisms that allow to match heteroclinic networks with the statistics of dominance times rely on the fact that their trajectories are characterized by spending long periods in neighbourhoods of saddle points from which they escape thanks to noise. In fact, in all the bistable perception models, noise plays a leading role to explain the statistics of dominance times of percepts observed in experiments. Noise is meant to model a diversity of inputs impinging on the areas represented in the model. Despite of being such a cornerstone, including noise is somehow a vague form to account for generic inputs since it considers inputs spanning a continuum of frequencies (the spectrum of the noise); it is, then, natural to think that perception transitions could be driven by less sophisticated inputs. With this idea, in this lecture we consider quasiperiodic perturbations of heteroclinic networks, assuming that our cognitive system (the featured areas of the model) is receiving events, either internal or from other brain areas, that include only a finite number of (non-resonant) frequencies. We show how these systems can achieve good agreement with Gamma distributions of the dominance times observed in bistable perception, and we compare these results with those obtained with noise. Moreover, we have adapted the well-known tool of separatrix map to the quasiperiodically perturbed heteroclinic networks. We then substitute the numerical integration by an iteration of maps which, at the end, provides a new (discrete) model for bistable perception. In addition, this approach allows to avoid numerical unstability when integrating close to saddle points.

This is joint work with Amadeu Delshams and Gemma Huguet (UPC).

Oscillations occur in virtually all spatial and temporal scales, performing crucial roles in biological systems. Yet, their precise function is often elusive and even dismissed as epiphenomenon. My work comprises mathematical and computational approaches to three oscillatory systems with distinct functional hypotheses: (1) inter-oscillator coupling provides response to seasonality in circadian systems (2) circadian phase clustering in the hippocampus underlie functions in memory processes (3) membrane potential oscillations coordinate cell polarity, growth and guidance in pollen tubes (the focus of this talk). Among the fastest growing cells in nature, pollen tubes sustain a highly polarized apical growth and are able to follow cues towards the ovule to perform fertilization. They show remarkable spontaneous oscillations in vitro involving multiple cellular processes such as growth, extracellular ion fluxes, and cytosolic ion concentrations. However, their function and underlying mechanism remain unknown, being limited by the availability of quantitative methods to characterize and model the different dynamic regimes observed. I will present a ‘Computational Heuristics for Understanding Kymographs and aNalysis of Oscillations Relying on Regression and Improved Statistics’, or CHUKNORRIS, which allowed to detect and quantify spiking behavior upon growth arrest and synchronized oscillations with growth and all ions studied (Ca2+, Cl- and H+). Conductance based models will be developed to predict the effect of these different regimes in polarity establishment, growth and guidance.

Mathematical biology and pharmacology models are increasingly utilized in the pharmaceutical industry, recognizing the need for improving the probability of success or reducing the cost of drug development. More mechanistic, quantitative systems pharmacology (QSP) models are being leveraged to aid in the identification of novel targets in early research, in the translational medicine activities for bringing molecules into the clinic, and for achieving proof of mechanism, and understanding variability in response to novel compounds in later clinical development.

During forward swimming, pairs of crayfish swimmerets (limbs) exhibit a robust pattern, moving rhythmically in a back to front metachronal wave with neighbors delayed by approximately 25% of the period. We study the mechanism responsible for this coordinated limb behavior using a model which represents the underlying neural circuitry as a chain of coupled oscillators. Previous modeling efforts have only considered the effects of nearest neighbor coupling, ignoring the presence of longer range connections in the system.

In this talk, I'll address how long-range coupling affects this mechanism, using an oscillator chain whose architecture reflects the known neural circuitry in the swimmeret system. I'll present analytical arguments and numerical simulations that indicate that long-range coupling tends to speed up the metachronal wave. Combined with insights from a computational fluid dynamics model, we suggest that this coupling may ensure maximal swimming efficiency.

In addition to the widely-used ability to selectively target specific cell types, Optogenetics and other emerging strategies for remote stimulation also offer an exciting path towards spatio-temporally-controlled targeting. For example, projected patterns of light can be used to selectively and flexibly control or image activity patterns distributed across entire populations of neurons.

The talk will present an overview of our recent work on distributed neuronal interfacing with large populations of neurons using optical and acoustic approaches. Our results demonstrate that patterned computer-generated holographic stimulation can achieve high temporal precision at multiple different scales, and can be effectively coupled with neuro-imaging and behaving animal paradigms. I will then describe a rational approach to quantitative understanding and design of next-generation neural interfaces using novel biophysical models, particularly the emerging Neural Intramembrane Cavitation framework for elucidating the biophysical basis of neuro-acoustic excitation. I will how insights provided from mathematical modeling may be opening new windows towards noninvasive neuromodulation with potential basic science and medical applications.

ABSTRACT