In this talk I will be presenting two topics: (1) studying conducting and selectivity functions of ion channels and (2) exploring tumor growth under cancer-immune system interactions and therapeutic treatments. These complicated biological systems usually consists of composite materials and involve intensive interactions among components. The former is at molecular level: multiscale treatments (atomic and continuum) and multiphysics (classical and quantum) are applied to different components (water molecules, channel proteins, membranes, and mobile ions, etc) according to biological importances of objects and computational efficiency. The solute-solvent surface serves as a free boundary to couple the discrete and continuum scales. The cancer research is macroscopic: interactions pathways of a large amounts of cells and cytokines are outlined from experimental observations and modeled by a systems of governing equations; the tumor is described as a moving domain with free boundary and has obviously different phase-field from normal tissues.
Both of the two topics require advanced numerical techniques for solving partial differential equations and simulations are validated by experimental data. Analysis, such as existence and uniqueness of the solutions are performed to these free boundary problems.
Task switching has been used as an experimental index of limitations on human cognitive flexibility. When switching from one task to another, participants exhibit increased reaction time, even when given ample time to prepare for the switch. Classical explanations of this "residual switch cost" typically presuppose that participants are fully motivated to perform the experimental task. This leads to interpretations of the residual switch cost in terms of structural cognitive factors. More recently, alternative explanations have proposed a role for motivational factors, arguing that preparation for task switches requires effort that participants are typically unwilling to expend, leading to only partial preparation (and residual switch costs) on average. In order to formalize the competing motivational and structural hypotheses of the residual switch cost into two alternative models, we employ a formal rational analysis of task switching experiments. This analysis shows that it is difficult to adjudicate between motivation and structural interpretations given existing data. We then conduct two new experiments to more conclusively test these ideas. Both experiments provide evidence for motivation-insensitive preparatory processes. Overall, this work casts doubt on the motivational interpretations of the residual switch cost, and provides a rigorous and principled framework for specifying and testing future hypotheses about motivational effects in the task switching paradigm.
Lung cancer is the leading cause of cancer-related deaths worldwide. Lack of early detection and the limited options for targeted therapies are both contributing factors to the dismal statistics observed in lung cancer. Thus, advances in both of these areas are likely to lead to improved outcomes. MicroRNAs (miRs or miRNAs) represent a class of non-coding RNAs that have the capacity for gene regulation and may serve as both diagnostic and prognostic biomarkers in lung cancer. Abnormal expression patterns for several miRNAs have been identified in lung cancers. Specifically, both let-7 and miR-9 are deregulated in both lung cancers and other solid malignancies. In this paper, we construct a mathematical model that integrates let-7 and miR-9 expression into a signaling pathway to generate an in silico model for the process of epithelial mesenchymal transition(EMT). Simulations of the model demonstrate that EGFR and Ras mutations in non-small cell lung cancers (NSCLC), which lead to the process of EMT, result in miR-9 upregulation and let-7 suppression, and this process is somewhat robust against random input into miR-9 and more strongly robust against random input into let-7.
In total variation denoising, one attempts to enhance an image by solving a constrained minimization problem with a total variation objective. The method has proven very effective at smoothing functions with discontinuities. More recently, it was also shown that such a model is capable of preserving regularity of the data. In this talk, I will present a convergent Rayleigh-Ritz method for approximating the smooth solutions of the L^2 total variation denoising model. The proof exploits the properties of functions of bounded variation, the approximation power of spline functions, and the non-expansiveness of the TV-denoising operator.
Systems biology aims to explain how a biological system functions by investigating the interactions of its individual components from a systems perspective. Modeling is a vital tool as it helps to elucidate the underlying mechanisms of the system. Many discrete model types can be translated into the framework of polynomial dynamical systems (PDS), that is, time- and state-discrete dynamical systems over a finite field where the transition function for each variable is given as a polynomial. This allows for using a range of theoretical and computational tools from computer algebra, which results in a powerful computational engine for model construction, parameter estimation, and analysis methods. In this talk, we show how a mathematical model helped us to inhibit tumor growth in melanoma cells.
We consider a two-species reaction-diffusion-advection competition model in which the species have the same population dynamics but different dispersal strategies. Regarding the advection coefficients as movement strategies of species, we investigate their course of evolution in the game-theoretical setting. By applying invasion analysis we find that if the spatial environmental variation is less than a critical value, there is a unique evolutionarily singular strategy, which is also evolutionarily stable. If the spatial environmental variation exceeds the critical value, there can be at least three evolutionarily singular strategies, one of which is not evolutionarily stable. Our results suggest that the evolution of directed movement of organisms depends upon the spatial heterogeneity of the environment in a subtle way.
Parasites are ubiquitous in nature, and the various biotic and abiotic processes that shape host population dynamics can also affect the host-parasite interaction. In this talk I'll discuss the dynamic consequences of such processes in three examples. First, I'll briefly discuss a project motivated by Mycoplasma gallisepticum infections in the House Finch, in which we consider virulence evolution under two trade-offs: the classic virulence-transmission tradeoff as well as a host movement tradeoff. Second, I'll briefly discuss the three-species dynamics of Daphnia (aquatic invertebrates), their parasites, and their algal food source, and I'll explain how analyzing the dynamics such a system has helped us clarify how certain biological phenomena drive some of the more interesting dynamics that can arise in this and other three-species models. Third, I'll spend the second half of the talk discussing more recent work modelling fish movement and population dynamics in response to changes in their physical environment (water temperature and dissolved oxygen [DO] levels), focusing on the population consequences of seasonal hypoxia in Lake Erie. I'll present results from a spatially explicit model that incorporates fish bioenergetics and nearly two decades of temperature and DO data from Lake Erie which we are using to explore how seasonal hypoxia impacts these populations, and I'll briefly discuss ongoing work using an extension of this fish movement model to assess how seasonal hypoxia will affect infectious disease transmission among fish.
A granuloma is a collection of immune cells that contains bacteria or other foreign material. An example is provided by the granulomas of Mycobacterium tuberculosis, a bacteria that infects a third of the world’s population. Although 90% of Tuberculosis cases are latent, 10% result in active infection. I will present a simple model of a generic granuloma and discuss efforts to discover why granulomas breakdown to cause active infections.
The time it takes a cell to divide, or intermitotic time (IMT), is highly variable, even under homogeneous environmental conditions. I will present a multistep stochastic model of the cell cycle and discuss how the model can be used to explain variability in IMT distributions and study the effect of drug treatment.
A stochastic interpretation of spontaneous action potential initiation is developed for the Morris-Lecar equations. Initiation of a spontaneous action potential can be interpreted as the escape from one of the wells of a double well potential, and we develop an asymptotic approximation of the mean exit time using a recently-developed quasi-stationary perturbation method. Using the fact that the activating ionic channel's random openings and closings are fast relative to other processes, we derive an accurate estimate for the mean time to fire an action potential (MFT), which is valid for a below-threshold applied current. Previous studies have found that for above-threshold applied current, where there is only a single stable fixed point, a diffusion approximation can be used. We also explore why different diffusion approximation techniques fail to estimate the MFT.
Shape-based regularization has proven to be a useful method for delineating objects from noisy images encountered in many applications when one has prior knowledge of the shape of the targeted object. When a collection of possible shapes is available, the specification of a shape prior using kernel density estimation is a natural technique. This process transforms the problem of shape-regularized image segmentation into an optimization problem involving a nonlinear energy functional.
In this talk I will present a framework for minimization of this energy functional with application to segmentation of still images. I will then present an extension of this approach to the identification of traveling fronts in image sequences.
Somitogenesis is a process for the development of somites which are transient, segmental structure that lies along the anterior-posterior axis (AP axis) of vertebrate embryos. The pattern of somites is governed by the segmentation clock and its timing is controlled by the clock genes which undergo synchronous oscillation over adjacent cells in the posterior presomitic mesoderm (PSM), oscillation slowing down and traveling wave pattern in the traveling wave region, and the oscillation-arrested in the anterior PSM, called determined region.
In this talk, I will focus on mathematical models which depict the kinetics of the zebrafish segmentation clock genes subject to direct autorepression by their own products under time delay, and cell-to-cell interaction through Delta-Notch signaling. First, for a basic two-cell system with delays, a sequential-contracting technique is employed to derive the global convergence to the equilibrium. This scenario corresponds to the oscillation-arrested for the cells in the determined region. Applying the delay Hopf bifurcation theory, the center manifold theorem, and the normal form method, we derive the criteria for the existence of stable synchronous oscillations for the cells in the tail bud of the PSM. These analytical results can be extended to a specific N-cell model. Hence, we provide an explanation for how synchronous oscillations are generated for the cells in the posterior PSM and how oscillations are arrested for the cells in the anterior PSM. Based on these results of two-cell system, we further construct a non-autonomous lattice delayed system to generate synchronous oscillation, traveling wave pattern, oscillation slowing-down, and oscillation-arrested in each corresponding region in the embryo to fit the experimental observations.
Frequency selectivity in the form of mode locking has been shown in stimulated nervous systems for various functional features. To understand the mechanisms behind these features, spiking neuron models have been used to study the precise timing of firing events thought to underlie frequency mode locking.
A number of neuron models have been developed and investigated in producing a large repertoire of neuronal behaviors, in attempts to develop an understanding of relevant functional features. We study a variant of the increasingly common use of the adaptive exponential leaky integrate-and-fire model, and explore mode locked solutions where the neuron is periodically stimulated. We present the analysis of mode locked solutions and their stability, and show numerical results demonstrating our analysis.
Binocular rivalry is the alternation in visual perception that can occur when the two eyes are presented with different images. Hugh Wilson proposed a class of neuronal network models that generalize rivalry to multiple competing patterns. The networks are assumed to have learned several patterns, and rivalry is identified with time periodic states that have periods of dominance of different patterns. In this talk we will use the theory of coupled cell systems to identify conditions under which networks with two learned patterns reduce to certain recent models of binocular rivalry where much of the dynamics are organized by a Takens-Bogdanov singularity. We also show that Wilson networks support patterns that were not learned, which we call derived. This is important because there is evidence for perception of derived patterns during several binocular rivalry experiments in the literature. We construct Wilson networks for these experiments and use symmetry breaking to make predictions regarding states that a subject might perceive.
An important feature of locomotion in cats, rats, and humans is that changes in speed occur due to a shortening of the stance (extensor) phase, while the swing (flexor) phase duration remains relatively constant. We have analyzed a simplified locomotor model that can replicate this key feature through feedback control. In this model, a central pattern generator (CPG) establishes a rhythm and controls the activity of a pendular limb, with afferent feedback signals closing the loop. Using dynamical systems methods, we analyze the mechanisms responsible for rhythm generation in the CPG, both in the presence and absence of feedback. We exploit our observations to construct a reduced model that is qualitatively similar to the original but tractable for rigorous discussion. We prove the existence of a locomotor cycle in this reduced system using a novel version of the Melnikov function, adapted for discontinuous systems. Finally we utilize our understanding of the model dynamics to explain its performance under various modifications, including recovery of oscillatory behavior after spinal cord injury and response to changes in load.
Directed cell migration, where cells migrate up/down chemical (e. g. chemokines, growth factors) or physical (e. g. stress, temperature) gradients, play important roles in a number of physiological processes; they include immune responses, tissue formation and cancer metastasis. In this talk, I will present work in my lab in understanding biophysical and biochemical mechanisms that cells use to migrate when subject to single or dual chemical gradients using an integrated experimental and theoretical modeling approach. Two examples will be given. First, I will describe how bacteria can sense chemical concentration gradients at a logarithmic scale; similar to sensory systems in high organism, such as human hearing and vision. I will also talk about how bacteria make movement decisions when subject to competing chemical gradients. Second, I will discuss about the roles of receptor-ligand binding kinetics in immune and cancer cell migration, and their implications in cancer metastasis.
Vertebrate seed dispersers and seed predators, insect seed predators, and pathogens are known to influence plant survival, population dynamics, and species distributions. My research investigates the importance of these groups of organisms in the sequential stages of early plant recruitment (i.e. from fruit developing in the crown to seedlings on the ground) in Neotropical forests. I use a combination of experimental and theoretical studies to investigate the influence of vertebrates, insects, and pathogens on plant survival and spatial patterns and the role of plant traits in mediating these interactions.
We simulate cellular automaton with a set of individual-based rules to reproduce spatiotem- poral dynamics arising from local interactions of ants in colony. Ants deposit diffusible chemical pheromone that modifies local environment for succeeding passages. Individual ants, then, re- spond to conspecific/heterospecific pheromone gradients, for example, by altering their direction of motion, or switching tasks. We describe ant’s movement by reinforced random walk, and study patterns (foraging trails, territoriality, etc.) emerging from ‘microscopic’ interactions of individual ants. We derive macroscopic PDEs by considering continuum limits of the mechanistic microscopic dynamics, and compare the two models.
The Cdc42 GTPase plays a key role in cell polarization in budding yeast. Although previous studies in budding yeast suggested positive feedback loops whereby Cdc42 becomes polarized, these mechanisms do not include spatial cues, neglecting the normal patterns of budding. In this talk, we present a two-equation reaction-diffusion model of cell polarization with a general function form of positive feedback. In the first part, we perform linear stability analysis, in particular Turing stability analysis to the model to derive conditions of parameters for which cell polarity emerges without any spatial cue. In the second part, we combine live cell imaging and mathematical modeling to understand how diploid daughter cells establish polarity preferentially at the pole distal to the previous division site despite the presence of two landmark cues at distal pole and proximal pole. We report that both spatial landmarks and GTP hydrolysis of Cdc42 by Rga1 controls the robust Cdc42-GTP polarization in diploid daughter cells.
When experiencing an ambiguous sensory stimulus (e.g., the vase-faces image) subjects may report haphazard alternations (time scale, seconds) between the possible interpretations. Various dynamical models that implement neuronal competition with reciprocal inhibition between neuronal populations show alternations behaving as noisy oscillators or as bistable systems subject to noise-driven switching. Slow negative feedback, neuronal firing adaptation or synaptic depression, sets the basic time scale (seconds) for switching. A minimal statistical model based on alternating renewal processes (with durations described by gamma distributions) captures various aspects of the percept time series.
Three-dimensional (3D) chromatin organizations play an important role in transcription regulation and can be used to define chromatin signatures. There is a possibility of long-range (i.e. 3D) epigenetic controls instead of the usual paradigm of promoter-cis control mechanism in distal regulation of gene. Such newly discovered mechanism may reveal novel epigenetic biomarkers for distinguishing between different cell types, for example, normal cells vs. cancer cells.
The current state-of-the-art experimental technique in 3D architecture inference is Hi-C, which was evolved from the earliest experimental protocol, Chromosome Conformation Capture (3C). In this talk, I will propose a flexible random effect model for inference from Hi-C data on constructing 3D architecture of chromatin. The model has advantages over conventional models thanks to its capability of incorporating both correlation structure and unknown sources of error into the model. Properties of the model will be presented, which will be followed by numerical simulations and applications to Hi-C human genome data.
A fundamental challenge in neuroscience is to connect behavior to the underlying neural mechanisms. Networks that produce rhythmic motor behaviors, such as locomotion, provide important model systems to address this problem. A particularly good model for this purpose is the neural circuit that coordinates limb movements in the crayfish swimmeret system. During forward swimming, rhythmic movements of limbs on different segments of the crayfish abdomen progress from back to front with the same period but neighboring limbs are phase-lagged by 25% of the period. This coordination of limb movements is maintained over a wide range of frequency. We examine different biologically plausible network topologies of the underlying neural circuit and show that phase constant rhythms of 0%, 25%, 50% or 75% phase-lags can be robustly produced. In doing so, we obtain necessary conditions on the network connectivity for the crayfish’s natural stroke pattern with 25% phase-lags. We then construct a computational fluid dynamics model and show that the natural 25% back-to-front phase constant rhythm is the most efficient stroke pattern for swimming. Our results suggest that the particular network topology in the neural circuit of the crayfish swimmeret system is likely the result of evolution in favor of more effective and efficient swimming.
Neural field models with transmission delay may be cast as abstract delay differential equations (DDE). The theory of dual semigroups (also called sun-star calculus) provides a natural framework for the analysis of a broad class of delay equations, among which DDE. In particular, it may be used advantageously for the investigation of stability and bifurcation of steady states. After introducing the neural field model in its basic functional analytic setting and discussing its spectral properties, we elaborate extensively an example and derive a characteristic equation. Under certain conditions the associated equilibrium may destabilise in a Hopf bifurcation. Furthermore, two Hopf curves may intersect in a double Hopf point in a two-dimensional parameter space. We provide general formulas for the corresponding critical normal form coefficients, evaluate these numerically and interpret the results.
The hippocampus plays an important role in representing space (for spatial navigation) and time (for episodic memory). Spatial representation of the environment is pivotal for navigation in rodents and primates. Two types of maps, topographical and topological, may be used for spatial representation. Rodent hippocampal place cells exhibit spatially-selective firing patterns in an environment that can be decoded to determine the animal’s location, heading, and past and future trajectory. We recorded ensembles of hippocampal neurons as rodents freely foraged in one and two-dimensional spatial environments, and we used a ``decode-to-uncover'' strategy to examine the temporally structured patterns embedded in the ensemble spiking activity in the absence of observed spatial correlates during rodent navigation. Specifically, the spatial environment was represented by a finite discrete state space.
Trajectories across spatial locations (``states'') were associated with consistent hippocampal ensemble spiking patterns, which were characterized by a state transition matrix of a hidden Markov model. From this state transition matrix, we inferred a topology graph that defined the connectivity in the state space. In contrast to a topographic code, our results support the efficiency of topological coding in the presence of sparse sample size and fuzzy space mapping.
The notion of excitability was first introduced in an attempt to understand firing properties of neurons. It was Alan Hodgkin who identified three basic types (classes) of excitable axons (integrator, resonator and differentiator) distinguished by their different responses to injected steps of currents of various amplitudes.
Pioneered by Rinzel and Ermentrout, bifurcation theory explains repetitive (tonic) firing patterns for adequate steady inputs in integrator (type I) and resonator (type II) neuronal models. In contrast, the dynamic behavior of differentiator (type III) neurons cannot be explained by standard dynamical systems theory. This third type of excitable neuron encodes a dynamic change in the input and leads naturally to a transient response of the neuron.
In this talk, I will show that "canards" - peculiar mathematical creatures - are well suited to explain the nature of transient responses of neurons due to dynamic (smooth) inputs. I will apply this geometric theory to a simple driven FitzHugh-Nagumo/Morris-Lecar type neural model and to a more complicated neural model that describes paradoxical excitation due to propofol anesthesia.
In this talk I will present some methods for constructing coupled dynamical systems out of simple bistable units that allow one to realise arbitrary finite state computational systems using coupled ordinary differential equations. More precisely, suppose one has an arbitrary strongly connected finite directed graph. How does one construct a systems of coupled cells that realises this graphs as attracting heteroclinic networks, with minimal limitations? The constructions are robust with respect to a suitably constrained set of perturbations and to addition of noise. On the presence of noise, there may or may not be short-term memory effects associated with the previous path on the network. This is joint work with C. Postlethwaite (Auckland).
Biological neural circuits display both spontaneous asynchronous activity, and complex, yet ordered activity while actively responding to input. When can model neural networks demonstrate both regimes? Recently, researchers have demonstrated this capability in large, recurrently connected neural networks, or “liquid state machines", with chaotic activity. We study the transition to chaos in a family of such networks, and use principal orthogonal decomposition (POD) techniques to provide a lower-dimensional description of network activity.
We find that key characteristics of this transition depend critically on whether a fundamental neurobiological constraint — that most neurons are either excitatory or inhibitory — is satisfied. Specifically, we find that constrained networks exhibit the transition to chaos at much higher coupling strengths than unconstrained networks. This property is the consequence of the fact that the constrained system may be described as a perturbation from a system with non-trivial symmetries. These symmetries imply the presence of both fixed points and periodic orbits that continue to act as an organizing center for solutions, even for large perturbations. In comparison, spectral characteristics of the network coupling matrix are relatively uninformative about the behavior of the constrained system.
Sensory integration and sensory binding are similar problems separated by a vast methodological gulf. The dominant paradigm of binding theory is neural synchronization, while sensory integration is built on observations of bimodal neurons. These cells show large increases in firing rates for bimodal presentation of weak stimuli, but little improvement for strong stimuli, a finding known as the Principle of Inverse Enhancement. It would be useful to link these two fields so that methods from each could be used by the other. The best case for such a bridge is the rattlesnake, which has two dissimilar visual systems, one for light and one for heat. Although this sounds like a binding problem, the rattlesnake has been studied using the methods of sensory integration. Many cells in rattlesnake optic tectum are sensitive only to light but can be strongly modulated by heat stimuli, or vice versa. I simulated these cells by assuming that they are members of synchronized pairs of excitatory-coupled rate-coded neurons. I replaced the usual weak coupling assumption with Goldilocks coupling: coupling is kept as strong as possible without distorting spike amplitudes. Both assumptions are unconventional but not unjustifiable. The same synchronized neuron model, without any parameter changes, accounts for a population of cells in cat visual cortex whose firing rates are enhanced by auditory stimuli. It also produces enhancements quite similar to those described psychophysically in humans and could be used to model some human color vision transformations; I present a model of the mysterious enhancement of "yellowness" generated from oscillatory synchronization of known neural mechanisms.
The predictability of neuronal network dynamics is a central question in neuroscience. First, we present a numerical investigation of the network dynamics of coupled Hodgkin-Huxley (HH) neurons and show that there is a chaotic dynamical regime indicated by a positive largest Lyapunov exponent. In this regime, there is no numerical convergence of the solution and only statistical quantifications are reliable. Second, we introduce an efficient library-based numerical method for simulating HH neuronal networks. Our pre-computed high resolution data library can allow us to avoid resolving the spikes in detail and to evolve the HH neuron equations using much larger time steps than the typical ones used in standard methods. Meanwhile, we can achieve comparable resolution in statistical quantifications of the network activity. Finally, we present a coarse-grained event tree analysis for effectively discriminating small differences in inputs to the network dynamics.
When observers view for extended time an ambiguous visual scene with two or more different interpretations they report switching between different perceptions. We focus on a classical paradigmatic stimulus, the visual plaids, consisting of two superimposed drifting gratings with transparent intersections (Wallach '35, Hupe & Rubin '03). For visual plaids, tristable perception is experienced: one coherent percept (the gratings move together as a single pattern) and two transparent percepts (the gratings slide across one another) with alternating depth order. In order to decipher the complex mechanisms of tristable perception, we gathered a large amount of psychophysical data on tristable plaids and developed a neural network, firing rate model of interaction between neural populations. The model developed can account for the dynamical properties (transition probabilities, distributions of percept time durations, etc) observed in the experiments and predicts that adaptation is strongly involved in perceptual switching.
The malaria parasite life cycle involves three cycles the sporogony (mosquito stage), exo-erythrocytic schizogony (liver stage), and the erythrocytic schizogony (human blood stage). We consider a simpli?ed mathematical model for malaria involving two parasite life cycles within the host namely the exo-erythrocytic and erythrocytic cycles. This study has revealed parasite replication characteristics which offer insights into the processes that allow the parasite to evade the human response during the red blood stages. First, the infection of the red blood cells by extracellular parasites during the erythrocytic cycle is characterized a reproduction number, R 0p ; that is less than one. Secondly, the asexual repro- duction of parasite during the red blood stage characterized by a reproduction number, R0m > 1; is responsible for the pathology of clinical malaria. Thirdly, we have found that the parasite depends mainly on the death of infected red blood cells to rapidly increase its population. Specifcally, the number of parasites, n1; in an infected red blood cell that dies need not be high for the parasite population to grow rapidly. We have found that for 8 ≤ n1 < 16; R0 > 1 and the parasite establishes itself while for 16 ≤ n1≤ 32; R0 < 1 and the parasite fails to establish itself. We are led to conclude that the parasite has preference for infecting older red blood cells as a strategy for evading the immune system.