Mathematical Biology Seminar - Fall 2017

The problem of tracking hidden states in nonlinear dynamical systems is confronted in a variety of contexts, particularly biology and neuroscience – tracking intracellular ion concentrations from neuron recordings, for example. For linear model systems, the optimal estimate is given exactly by the Kalman filter, but nonlinear systems demand either linearizing approximations or statistical representations such as Monte Carlo sampling. I will describe an alternative approach to optimal estimation that leverages a suggestive correspondence with statistical path integrals. Among these are an iterative homotopy continuation method for tracking dominant contributions to this integral, variational annealing, as well as a formulation in a Hamiltonian manifold, where artificial momentum directions permit a fuller search over the model dimensions. I will illustrate the applicability and reliability of these estimation methods in both chaotic toy models and biophysical neuron models.

The nervous system is a sophisticated control system, riding at the helm of an equally sophisticated "plant". Understanding how the nervous system encodes and processes sensory information and computes motor action, therefore, involves understanding a complex closed-loop control system. This talk will present a research program devoted to developing and applying ideas from engineering to decode closed-loop control in animals (including humans). In addition, the talk describes how new insights into biological control systems can, itself, feed back to engineering by providing new ideas for mechanisms and control systems in engineering.

Numerical techniques have predicted that reentrant electrical scroll waves underlie many cardiac arrhythmias, but experimental limitations have hampered a detailed understanding of the specific mechanisms responsible for reentrant wave formation and breakup. To further this effort, we recently have begun to apply the technique of data assimilation, widely used in weather forecasting, to reconstruct time series in cardiac tissue. Here we use model-generated synthetic observations from a numerical experiment to evaluate the performance of the ensemble Kalman filter in reconstructing such time series for a discordant alternans state in one spatial dimension and for scroll waves in three dimensions. We show that our approach is able to recover time series of both observed and unobserved variables that match the truth. Where nearby observations are available, the error is reduced below the synthetic observation error, with a smaller reduction with increased distance from observations. Using one-dimensional cases, we provide a deeper analysis showing that limitations in model formulation, including incorrect parameter values and undescribed spatial heterogeneity, can be managed appropriately and that some parameter values can be estimated directly as part of the data assimilation process. Our findings demonstrate that state reconstruction for spatiotemporally complex cardiac electrical dynamics is possible and has the potential for successful application to real experimental data.

Over the past 40 years the cost of 1 gigaflop per second of computing capability has decreased by nine orders of magnitude. As a result, highly detailed biological models can now be simulated with minimal computational cost. Unfortunately, during that same time frame, the abilities of our analytical tools to understand the complicated mechanistic behavior of these models have not evolved at the same pace. My particular focus has been on the creation and utilization of mathematical tools which can reduce the dimensionality and complexity of differential equations describing nonlinear biological systems, making them amenable for further study.

In this talk, particular attention will be given to the problem of the elimination of cardiac alternans, a pathological beat-to-beat alternation of electrochemical cardiac dynamics which is widely regarded as a precursor to cardiac arrest, a leading cause of death in the industrialized world. Using a recently developed method of isostable reduction, both single-cell and partial differential equation models of cardiac behavior are investigated in order to formulate an energy-optimal control strategy for the elimination of alternans. Additionally the effect of stochastic pacing on the genesis of discordant alternans will be discussed to posit a mechanism by which depressed heart rate variability increases one’s risk of sudden cardiac death.

Electromagnetic recordings of the brain that show transitions from high amplitude to low amplitude signals are likely caused by an underlying changes in the synchrony of neuronal population firing patterns. A classic example is the event-related oscillatory phenomenon known as post-movement beta-rebound (PMBR), where a sharp increase in EEG or MEG power is seen at beta frequency following movement termination. A related phenomenon is movement related beta decrease (MRBD), whereby beta rhythms are suppressed during movement.

Traditionally neural mass models have been used to model large-scale brain dynamics, however they fail to account for the degree of synchronisation within a population. This could be tracked within a large-scale model of synaptically interacting conductance based neurons, though at the expense of analytical tractability. Thus it is of interest to seek levels of description that provide a bridge between microscopic single neuron dynamics and coarse grained neural mass models, while preserving some notion of within-population coherence.

I will present a parsimonious model for the dynamics of synchrony within a synaptically coupled spiking network that can replicate a human MEG power spectrogram showing the evolution from MRBD to PMBR. Importantly the high-dimensional spiking model has an exact mean field description that allows considerable insight into the cause of beta-rebound. Interestingly the reduced model takes the form of a generalised neural mass model where the standard sigmoidal firing rate has been replaced by a derived quantity that is a function of the Kuramoto order parameter for synchrony.

Today I will discuss both modeling of biological neuronal networks and a method to test such models. First I will describe a small-scale model of nucleus HVC of the zebra finch, a region implicated in song control. The core element of the model is a functional architecture that can exhibit multiple modes of activity depending on model parameter values. Specifically, the strengths of inhibitory chemical synapses modulate a pattern of activity among excitatory neurons. The model reproduces many observed features of HVC population activity.

Next I will describe an optimization technique for determining which measurements are required to estimate unknown model parameters. Specifically, I will use simulated time series of membrane voltages to estimate the synaptic strengths and electrophysiological properties of the neurons in the HVC model, where the test of success is the ability to predict the associated mode of network activity. In addition, I will show how the procedure can "prune" a model to the maximum dimensionality required to capture observations. Finally, I will comment on plans to approach the (formidably challenging) case in which the measurements consist of real biological data.