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Major Findings from Recent Research Activities (2004-2005)
Neural rhythms: biophysics and dynamics
Working with White and Nancy Kopell, Theoden Netoff has published two papers (Netoff et al. 2005a, b) on neuronal synchronization. These papers involve experimental application of phase response techniques. In new findings, they have estimated the so-called �infinitesimal� phase response relationship under different experimental conditions (scaled inputs, different firing rates, Poisson-process-driven barrages of inputs). The results show that physiologically realistic inputs can be treated validly as �weak,� implying that a powerful set of mathematical tools may be applied to this situation. Previously described modeling work by Netoff and Clewley, working with White, has been published (Netoff et al. 2004)
Dmitri Pervouchine and Theodon Netoff, along with Research Asst. Prof. Horatio Rotstein, continued work with Kopell and White on a paper concerning gamma (30-80 Hz), beta (12-30 Hz) and theta (4-12 Hz) rhythms in the entorhinal cortex (EC), the gateway between the hippocampus and the neocortex. This paper considers modules of several kinds of cells (fast-spiking (FS) interneurons, stellate cells and pyramidal cells) having different intrinsic currents and neurotransmitters. They show how the dynamics can be reduced to low-dimensional maps, and use that to explain how different rhythms arise in the same module. The work builds on recently accepted work by Rotstein et al. (2005) on theta rhythms in the CA1 area of the hippocampus and two papers recently published by Netoff et al. (2005a,b) on the use of hybrid in vitro-in silico networks modeling some entorhinal cortex dynamics.
Jozsi Jalics, Tilman Kispersky and Nancy Kopell are using numerical and analytical techniques to continue to investigate the role of rhythms (theta, beta, gamma, and theta nested gamma) in the formation and coordination of neuronal ensembles in the superficial layers of the entorhinal cortex. Based on experimental evidence, they are working under the hypothesis that the EC has a modular structure in which each module consists of populations of pyramidal cells, interneurons, and stellate cells. They have shown that theta and gamma rhythms can interact with focal inputs to produce ensembles of modules. Also, they are focusing on the role of inputs from the medial septum, which acts as a theta generator, in the generation of theta nested gamma rhythms and ensemble formation. Depending on the relative strengths of the two components of the septal input, they have shown that theta-nested-gamma activity can be generated through the activation or suppression of interneuron firing in phase or out of phase with the septal theta while interneuron suppression can lead to disinhibition of the stellates, which tend to be suppressed by gamma frequency inhibition from the interneurons. Also, stellates participating in a theta rhythm in one module can modulate gamma activity in another module through theta frequency activation of the interneurons of the second module. Some of this work has been presented at the 2004 Society for Neuroscience meeting, and other parts have been submitted as an abstract to the 2005 meeting.
New work by Netoff and Tilman Kispersky focuses on imaging coherent activity from 10-100 cells simultaneously. For this work, they load individual neurons in the entorhinal cortex with fluorescent indicators of the intracellular calcium concentration. This work will be presented at the Society for Neuroscience meeting in Fall 2005. Work is planned with White and Kopell on the dynamics revealed by this technique.
Work in progress with Pervouchine, Kopell and the lab of Miles Whittington on a slow rhythm (< 1 Hz) in the entorhinal cortex makes use of metabolic activity as well as standard synaptic dynamics. The associated mathematics uses a combination of mean-field and single cell treatments. Further results have also been obtained in understanding how the active part of the slow rhythm produces both theta and beta rhythms, using both simulations and low dimensional maps.
Brian Burton is working with White on two projects. In the first project, Burton is studying how cellular oscillations develop in the entorhinal cortex around the time that rat pups� eyes open. Burton�s data show that the density of persistent Na+ channels increases during the crucial time period. Preliminary data suggest that the density of slow inward rectifying channels underlying the �H conductance� increase as well. In the next step, Burton will attempt to convert �young� cells into �old� cells using dynamic clamp techniques. In the second project, Burton is studying oscillatory entorhinal neurons using techniques from communication theory. His results show that cells respond preferentially to sequences of input that match the statistical properties of spike trains. In the future, Burton will focus on the mechanisms underlying these spike train statistics.
Horacio Rotstein, Tim Opperman (a visitor from the lab of Andreas Herz), John White and Nancy Kopell have recently submitted a paper on how subthreshold oscillations in the EC are formed, and what determines their time constant. The subthreshold behavior is shown to be related to �canard structures�, in which trajectories of equations spend significant time near an unstable manifold. This make use of ideas involving reduction of dimension, using the structure of the HH equation: they show that the interspike interval can be divided into subintervals inside which the relevant trajectories can be reduced to a two- or three-dimensional systems that are amenable to geometric or analytic study. In addition to variation in time scales, this reduction uses the fact that the some ionic currents are displayed at significant levels only during parts of the trajectory, leading to a natural break into subintervals. They also explain heuristically the effects of noise in creating more robust subthreshold oscillations. A related paper on reduction of dimensions by Clewley et al. involving some of these authors has recently been accepted. John White and Martin Wechselberger also participated, and work is in progress showing that the analysis of Wechselberger on 3-D canard structures is relevant to this work. In follow-up work with Opperman, Rotstein and Kopell use similar ideas to investigate how the subthreshold oscillations are related to resonance, how the resonance differs from that of linear systems, and the role played by individual ionic currents in producing it.
In related work on reduction of dimensions, Jozsi Jalics, Horacio Rotstein and Nancy Kopell are analyzing a biophysical model of an EC layer V pyramidal cell including a persistent sodium and slow potassium current using reduction of dimension techniques and bifurcation analysis to determine the mechanisms for the generation of subthreshold oscillations, mixed mode oscillations, resonance, and rebound firing as well as its synchronization properties and role in network activity, such as the generation and propagation of the theta rhythm. Interestingly, they have found that the spiking currents are important in the interspike interval (usually they are ignored), and that in the mixed mode oscillation regime the dynamics of the delayed rectifier potassium current plays an important role in the analysis of the 3-D canard structure. See the section on Pattern Formation for related work.
The work with Alexey Kuznetsov and Charlie Wilson on the dynamics of dopamine neurons continues to progress. A paper was submitted and is now in revision. See last year�s report for more details on this work.
Neural plasticity and single-neuron biophysics
Yu-Dong Zhou is continuing to work with White to understand the mechanistic basis of spike-time-dependent plasticity (STDP), a phenomenon by which synaptic weights change dynamically in some neuronal networks. In previous years, Zhou demonstrated that STDP occurs in the entorhinal cortex and that STDP depends on postsynaptic spike width. More recent work by Zhou has shown directly that STDP depends on the form of the postsynaptic calcium transient; that STDP requires NMDA receptors; and that calcium transients through dendritic voltage-gated calcium channels modulate STDP. This work suggests that the almost universally accepted model of STDP induction requires modification. Acker�s modeling work suggests the form of this modification. This work was presented at the Society for Neuroscience meeting in Fall 2004 and will be submitted for publication in Summer 2005. In future work, Zhou will work with White and Kopell to understand the cellular mechanisms of population oscillations in brain slices of entorhinal cortex. The work of Corey Acker, mentored by White and Sen, focuses on mechanistic computational models of spike-time-dependent plasticity (STDP). This work complements the experimental studies of Zhou cited above, and will be submitted in summer 2005.
Corey Acker is also studying back-propagating action potentials in computational models. His technique for this problem, developed with assistance from Wayne, allows Acker to describe back-propagating action potentials in morphologically complex structures in a pseudo-analytical fashion. He finds that channel densities and channel properties are the most important contributors to action potential propagation in pyramidal cell dendrites. Morphological factors are secondary in importance. This work will be submitted in summer 2005. Wayne is now working with Kopell and grad student Dorea Vierling-Classen to try to give a rigorous foundation to numerical simulations of Acker on traveling waves in neural models whose physical properties change as one moves along the neuron. They are examining analytically traveling wave solutions to FitzHugh-Nagumo under slowly varying parameters. These parameters are analogous to the changes in diameter, distribution of ion channels and variation in channel type along the length of the dendrite modeled by Acker.
Tara Keck has continued work with John White, focusing on how dendritic inputs and back-propagating action potentials (bAPs) interact. Previously, she found that properly timed bAPs reduce the efficacy of excitatory inputs. More recently Keck has made a number of new discoveries: (1) Synaptic efficacy is also strongly modulated by postsynaptic, but not presynaptic, spike rate. This appears to be an entirely novel form of short-term synaptic plasticity. (2) The mechanism of postsynaptic rate dependence of synaptic efficacy involves strychnine-sensitive glycine receptors. Under control conditions, these receptors are activated and appear to reduce synaptic efficacy by shunting the postsynaptic membrane. Bouts of high-frequency activity in postsynaptic neurons close the glycine receptors, apparently by up-regulating the activity of glycine transporters. (3) These mechanisms have clear effects on long-term synaptic plasticity as well. This work will be submitted for publication and defended in 2005.
Kyle Lillis, currently in his second year, is working with John White to put together a dissertations prospectus (proposal), to be defended in late 2005. Lillis plans to study the electrophysiological and biochemical signaling mechanisms involved in STDP. He will also extend the STDP-related modeling work of Acker.
Neural rhythms and behavior
Steven Epstein is continuing work with Christoph B�rgers and Nancy Kopell on the mechanisms and functions of cortical gamma oscillations in attention. Their investigations of a medium-scale cortical local-circuit model have shown that cholinergic modulation, by reducing adaptation currents in principal cells, induces a transition from asynchronous spontaneous activity to a "background" gamma rhythm in which individual principal cells participate infrequently and irregularly. In a paper just published (Borgers et al. 2005), they proposed that rhythms of this kind characterize states of preparatory attention and have demonstrated a novel mechanism by which gamma oscillations serve to increase stimulus sensitivity and enhance stimulus competition, computations fundamental to selective attention. The background dynamics facilitates the formation of cell assemblies, as well as suppressing competing (distractor) signals. The simulations use a reduction of the complex Traub model for persistent kainite-induced gamma in the neocortex and entorhinal cortex. The large Traub model includes multiple compartments for each cell, and electrical coupling between axons of cells to produce noise. In the reduced model, the noise is introduced artificially into 1-compartment neurons. Further work is in progress, concerning selective attention, rather than vigilance.
With the labs of Tengis Gloveli and Whittington, Rotstein and Kopell have been working on a paper (just submitted) concerning how the different kinds of tissue slices can produce different rhythms, depending on the angle at which the slice is made. Using anatomy and physiology, the paper shows that some angles produce gamma rhythms, others produce theta, and in-between angles produce a nesting of these rhythms. The differences are shown to be related to differences in projections of different classes of cells (O-LM cells, which produce theta rhythms, and pyramidal cells that, with FS- interneurons, produce gamma rhythms. Modeling shows that the two compete for control of the FS interneurons, and that angles of the slices affects which cells are dominant. This is important because the anatomy of the cells is related to classes of inputs to the cells. These results shed light on the how the gamma and theta rhythms help to produce and coordinate cell assemblies.
Postdoc Ehud Sivan has worked with Kopell on suggesting a mechanism and circuitry for odor analysis in insects. They hypothesized that two aspects of odor recognition, odor clustering and fine discrimination of odors, are encoded in parallel by two brain areas of the insect olfactory system: Population activity of neurons in the lateral horn (LH) encodes the odor cluster and population activity of neurons in the mushroom body (MB) encodes the fine identity of the odor. Their mechanism is based on their hypothesis that the underlying network of the insect olfactory system consists of a repetitive, hard wired, substructure. In a paper published in PNAS (Sivan and Kopell 2005) they showed that these suggested mechanisms and circuitry explain not only the observed numbers and connections of neurons in the system, but also the observed activity of these neurons, and why oscillations are critical for fine discrimination but not for clustering of odors.
Sivan and Kopell have also worked on modeling the oscillations and slow patterning in the activity of projection neurons (PNs) in the antennal lobe (AL) of insects seen when an odor is presented. Experimental results indicate that the oscillations are the result of the interaction between the PNs and the inhibitory local neurons (LNs) in the AL; e.g., application of picrotoxin (an antagonist of the inhibitory transmitter GABA_A) abolishes these oscillations. The slow patterning, on the other hand, was shown to be resistant to such blockage. In a Hodgkin-Huxley model, Sivan and Kopell reproduced both the oscillations and the slow patterning. Their results showed that, as previously suggested, the oscillations are the result of the interaction between the PNs and LNs. They also showed that calcium and calcium-dependent potassium channels (found in PNs of bees and moths) are sufficient to account for the slow patterning resistant to picrotoxin. The intrinsic bursting property of the PNs, resulting from these additional modeled currents, give rise to another network feature that was seen experimentally: a relatively small increase in the number of additional generated PN action potentials when LN input is blocked. Consequently, the major effect of network inhibition is to redistribute the action potentials of the PNs from bursting to one action potential per cycle of the oscillations. They hypothesize that the slow patterning is important for the segmentation of odor mixtures into components. A paper was recently submitted.
Research in the Hasselmo laboratory has focused on developing models of cortical mechanisms for memory-guided behavior. This includes modeling of performance of rats in tasks, including spatial alternation, spatial delayed non-match to position and odor list recognition and an operant visual recognition task in monkeys. Simulations guide the movement of the animal in an experimental task, using a prefrontal cortex network model for action selection based on memory retrieval from a hippocampal network simulation. This work has proceeded on a number of different levels, including detailed Hodgkin-Huxley models of region CA3 developed (Kunec, Hasselmo and Kopell 2005) which replicate spike timing relative to theta rhythm oscillations; integrate-and-fire models done in the CATACOMB simulation package which replicate prefrontal cortex spike timing during performance of an operant task in monkeys (Koene and Hasselmo, 2005a, 2005b) and the KINESS simulation package (Gorchetchnikov and Hasselmo, 2005; Gorchetchnikov et al., 2005); and more abstract models using threshold neurons in MATLAB (Hasselmo and Zilli, 2005; Hasselmo and Eichenbaum, 2005) that replicate spike timing relative to spatial location observed in rats during performance of spatial memory tasks.
Cecilia Diniz Behn is working with anesthesiologist/statistician Emery Brown, physician/experimentalist Thomas Scammell, and Kopell on modeling transitions between sleep and wakefulness. The network of neurons in the brainstem and hypothalamus that controls these transitions gives rise to interesting dynamics on multiple time scales. Mathematical techniques for reduction of dimension permit analysis of the model and provide insight into some of the mechanisms that may underlie the sleep-wake transitions, including micro-arousals. The work is aimed at understanding the response of normal animals to challenges such as sleep deprivation, as well as mechanisms involved in sleep disorders, such as narcolepsy and insomnia.
Michelle McCarthy is involved in ongoing work with Kopell and Emery Brown on the biophysical and network mechanisms involved in the increase in beta activity observed in the EEG in patients who are given low doses of propofol anesthesia. Low-dose propofol, a GABAa potentiator, induces behavioral excitation rather than sedation in patients. Using simulated populations of neurons with Hodgkin-Huxley-type dynamics, they find rhythmical changes in the population that are consistent with the EEG frequency changes seen with low-dose propofol. The population consists of pyramidal cells and interneurons interconnected synaptically. Intrinsic currents include the spiking sodium and potassium currents as well as two additional potassium currents, a slow potassium current and the A-current. The presence of the slow potassium current in the interneurons appears to be critical to the phenomenology of increased beta activity in the presence of slight GABAa potentiation .
Also see �Dynamics of the auditory system� below.
Control of neural dynamics
Jonathan Bettencourt has continued work with White and others in White�s lab to create the next generation of the lab�s real-time experimental control system. As discussed last year, this system allows the user to interact with neurophysiological or other preparations in real time, at clock rates of up to 50 kHz. The main purpose of the system is to test dynamical systems-inspired hypotheses in �wet� experiments. Bettencourt�s efforts the last year have focused on making the system more robust and straightforward to install and use. For example, he is developing a system that automatically runs from a CD, usable by an experimentalist with very little experience with computers. Bettencourt will defend his MS thesis in late 2005.
In academic year 2004-2005, Jonathan Sip worked with White and collaborator Howard Eichenbaum to develop inexpensive headstages for recording from specific locations in the brains of awake, behaving rats. These new headstages are superior to existing technology in cost and ease of use.
Dynamics of the auditory system
Rajiv Narayan has continued his work on discrimination of natural sounds by auditory neurons in the songbird auditory cortex analog field L. Narayan previously worked on a computational model of discrimination, where he related the structure of experimentally observed receptive fields to the discrimination of natural sounds such as birdsongs. He found that key parameters of the receptive field that produce efficient discrimination are the relative timing and the balance of excitatory and inhibitory regions in the receptive field. A manuscript based on this work is now under review at the Journal of Neurophysiology. Over the last year, Narayan has also performed electrophysiological recordings from field L of the songbird. The analysis of neural discrimination based on this new data is the focus of a new manuscript being prepared for submission.
Gilberto Grana has learned how to perform electrophysiological recordings in awake songbirds. Much of the work in the field has been performed on anesthetized birds. Grana�s goal is to compare the awake results to the anesthetized results to look for possible differences, e.g., in the structure of Receptive Fields. One of his main goals is to use the awake system to investigate short term and long-term plasticity in receptive fields. Grana has already obtained preliminary data from the awake experiments. These data will contribute to the manuscript in preparation on neural discrimination.
Gabriel Soto has been working jointly with Kopell and Sen on a model of layer IV in auditory cortex, investigating the relative roles of thalamocortical and intracortical circuits in shaping receptive field structure in auditory cortex. Soto�s model provides new insights into the relationship between the architecture of cortical networks and Receptive Fields. Using his model, he has identified key network parameters that may strongly influence RF structure, including the recurrent excitatory connections and excitatory to inhibitory connections. Soto has also been investigating how neuromodulators such as acetylcholine, a major neuromodulator of auditory cortex, affect the network behavior. He has found that acetylcholine shifts the model network into a different state that displays three important differences: the ratio of thalamic vs. cortical contribution to the response is increased, the frequency tuning of cortical cells is sharpened, and cortical cells become more synchronized, producing stronger gamma oscillations.
Gabriel Soto has also been working with Sen and Kopell on the role of rhythms in longer-term plasticity of the auditory pathway. Tone-shock pairing experiments not only induce behavioral responses to the conditioned stimulus (CS) frequency but also induce long term plasticity of cortical Receptive Field (RF) (Weinberger, Nat Rev Neurosci 2004), shifting the previous best frequency (BF) towards CS. These experiments suggest that frequency-specific thalamic inputs combine with non-specific thalamic inputs triggered by the shock and release of acetylcholine (ACh) by the Nucleus Basalis to induce long-term changes in RF. Non-specific thalamic stimulation also induces oscillations in the gamma-frequency range (fN) throughout ACx (Sukov and Barth, J. Neurophys 2001). This suggests that the interaction between thalamic inputs, encoding frequency-specific information and cortical cells undergoing gamma oscillations, elicited by the shock and ACh, could affect the strength of thalamocortical synapses. Soto, Sen and Kopell have shown that the local architecture of the cortical network and gamma oscillations within this network can trigger potentiation or depression of thalamocortical synapses according to spike-timing dependent plasticity. They have also found that potentiation or depression at the thalamocortical synapse depends on the relationship between fI, the firing rate of frequency-specific thalamic inputs and fN, the natural frequency of oscillations in the cortical network. This work will be presented at the 2005 Soc. for Neurosci. meeting.
David O�Gorman is working with White and Chris Shera of MGH, modeling the responses of auditory nerve fibers to electrical stimulation by cochlear implants. O�Gorman�s work suggests that the subthreshold dynamics of spiking axons are crucial for understanding how these axons respond using current stimulus protocols. O�Gorman�s work is leading to new paradigms for cochlear implant design.
Dorea Vierling-Claassen and Kopell have been part of a working group at BU on rhythms and schizophrenia with schizophrenia researchers from Mass. General Imaging Center and McLean Hospital. They are studying pathologies in the rhythmic electrical responses of the auditory cortex of persons with schizophrenia when given periodic stimuli. The modeling is used to understand how known deficits at the cellular and synaptic level, in particular alterations to cortical fast spiking interneurons, might affect the responses to auditory stimuli. The current focus is on data demonstrating pathologies in the gamma and beta rhythms during paradigms involving periodic auditory input.
Pattern formation and dimension reduction in nonlinear PDEs and ODEs
Nicolas Popovic worked with Wayne and Kaper on a new project to use dynamical systems theory (specifically the theory of invariant manifolds) to understand relaxation phenomena in hyperbolic conservation laws. In systems of hyperbolic partial differential equations that couple conservations laws with rate equations, there is often a relaxation toward an asymptotic state. They are attempting to understand this process by constructing invariant manifolds for this system of PDE's in which the motion on the manifold describes the limiting state and the approach to the manifold gives the relaxation toward this state. Since these systems often involve two different time scales (for the relaxation and reaction) the problem involves singular perturbations. They hope that the insights gained in applying geometric singular perturbation theory in various finite dimensional examples will be of use in this infinite-dimensional context.
Popovic and Kaper completed an article (submitted to the Journal of Dynamics and Differential Equations) on critical waves speeds for traveling waves (TWs) in reaction-diffusion equations. The class of equations includes the famous Fisher -Kolmogoroff equation with a quadratic nonlinearity and the classical bistable case with a cubic nonlinearity, as special subcases; here the order of the nonlinearity is considered as a parameter. The critical wave speed is the value of the wave speed at which the transition from an algebraically decaying TW to an exponentially decaying TW takes place. Popovic and Kaper identified a natural small parameter and employed geometric desingularization, or `blowup', to generate rigorous asymptotic expansions for the critical wave speeds. They also explained why one of the expansions is in fractional powers of the small parameter, a phenomenon that had previously been explained as being a necessity of asymptotic matching. The blowup method is a method from dynamical systems theory that Popovic used in his PhD thesis in Vienna and that Kaper used with former CBD student M. Hayes, PhD 1999.
Margaret Beck, working with Kaper and A. Doelman, has completed the existence proof for traveling waves in the Oya-Vallochi model of bioremediation. This model consists of three reaction-diffusion equations, one for each of the concentration of the substrate (harmful organic pollutant that is to be remediated), the concentration of an electron acceptor (that acts as a food supply for the in-situ bacteria), and the concentration of active bacteria. Beck showed that this model can be reduced to a fast-slow system of five ordinary differential equations in the traveling wave variable. Moreover, there is a three-dimensional normally-hyperbolic invariant manifold in the phase space of this fast-slow system, and Beck used geometric singular perturbation theory (GSPT) to construct the traveling wave in the transverse intersection of invariant manifolds. A number of technical obstacles concerning the nonuniform differentiability of the vector field had to be overcome before the GSPT could be applied. Moreover, her analysis identifies, for the first time, how the peak height of the traveling wave depends on each of the physical parameters, and excellent agreement is found between the theoretical predictions and the results of numerical simulations. Their article was submitted to the Journal of Nonlinear Science, and they are presently starting the stability analysis.
Kaper and Wayne are working with Beck to understand the stability properties of traveling waves in conservation laws like Burgers' equation. While the local stability of these traveling waves is understood through the work of D. Sattinger, T. Kapitula and others, they hope to construct finite dimensional invariant manifolds in the phase space of this equation (near the fixed point representing the traveling wave), which will give a very precise description of the asymptotic approach of solutions toward the wave. They are also exploring whether or not they can construct Lyapunov functionals for this equation, which would give more global results on the stability of such traveling waves.
Antonias Zagaris' first paper with Bill Gear (Princeton), Yannis Kevrekidis (Princeton), and Kaper, in which they presented the zero-derivative principle for reduction of complex systems from chemistry, is in press at the SIAM Journal on Applied Dynamical Systems. This paper was described in our report of the previous year. This year, they completed part 2 of this work, in which they establish the mathematical validity of the zero-derivative principle. For this proof, Zagaris developed an elegant bootstrapping method. This method of proof will help streamline the validity proofs for other algorithms. Zagaris continues to work with Kaper on this part-time since starting a postdoc position in Amsterdam, the Netherlands.
Zagaris built significantly on earlier work with Kaper and Kaper (published in Multiscale Modeling and Simulation). He developed a general framework for reduction methods of complex systems of ODEs by working in the tangent bundle. The equations for the species concentrations and the equations for the vector fields are simultaneously studied in the tangent bundle. This joint perspective is especially important for analyzing and further developing reduction methods, since some commonly- used reduction methods, such as the zero-derivative principle, are applied to the equations for the species concentrations, whereas other methods (such as the CSP method analyzed in their previous work) are applied in the spaces of the vector fields. They identify the reduction induced in the space of the vector field by a method that works in the space of the species concentrations, and also identify the induced reduction methods in the concentration space by a method applied in the space of the vector field. Zagaris's article is to appear in the journal Mathematische Nachrichten. Work in this direction is now focused on (i) extending the tangent bundle analysis to more general vector fields; (ii) extending the proof of validity of the CSP and ZDP methods from the realm of fast-slow systems to a larger class of ODEs in which there is a spectral gap; and (iii) streamlining the induction proof of validity of the CSP method by using the bootstrap method that they developed in the project described above. Other work by Zagaris is described in last year�s report.
Rotstein has been working with Rachel Kuske on localized and mixed-mode oscillations in a model of a coupled pair of calcium oscillators. In the former, one oscillator displays small-amplitude oscillations and the other displays large amplitude oscillations. In the latter, each oscillator alternates between large and small amplitude oscillations, which are not necessarily synchronized. This work differs from previous work with Kopell on localization in globally coupled BZ models, in which the localization is mainly due to the global coupling term; in the current work, the local coupling created by diffusion is the main source of localization. Follow up work on the BZ reaction is aimed at understanding the global effects of the global coupling on the coupled oscillators. In each of these cases, the localization is related to the formation of �canards�, small amplitude oscillations that can explode into large relaxation ones. The computations are based on the analysis of Krupa and Szmolyan, which allow local effects of the coupling near the Hopf bifurcation point to be explained in terms of each oscillator separately. The study of the global effects involves the development of new techniques.
A related project on canards involves Nicola Popovic as well as Rotstein, Kopell and Martin Wechselberger. The work continues previous work of Medvedev and Kopell, and more recent work of Medvedev on the so-called Wilson-Callaway model, which has been proposed to explain the various firing patterns observed in dopaminergic neurons in the mammalian brain stem, in the limit of strong electrical coupling. A characteristic feature of the model is the occurrence of mixed-mode oscillations (i.e., trajectories in which small amplitude and large amplitude oscillations occur in sequence) in a wide range of parameters. They use geometric singular perturbation theory to show that these oscillations are caused by a canard phenomenon occurring in a reduced set of equations obtained by a global center manifold reduction. The Medvedev paper gave a heuristic explanation for the mixed modes; the current work uses the analysis of Wechselberger to systematically identify the associated bifurcation (Farey) sequences.
Wayne has recently begun to work with Chelsea Smock. Smock has been reading papers on the stability of vortex solutions for the two and three-dimensional Navier-Stokes equations and they hope to extend these results to other fluid flows over this coming summer.
Oleg Mittichenko is working with Popovic, Wayne and Kaper on geometric desingularization via the power geometry method. The goal is to work out the connection between this approach and the blow-up technique.
Christophe Lecomte, working jointly with AME faculty Paul Barbone and J Greg McDaniel, derived a class of bounds on error introduced by projecting a high dimensional dynamical system into a low dimensional subspace. These led to a paper that was submitted to IJNME, and is currently being revised according to reviewer comments.
Gene Dynamics
Will Blake, Farren Isaacs, grad student Kevin Murphy and Jim Collins investigated the role that stochasticity, or noise, plays in the dynamics of eukaryotic gene regulation and expression, and the mechanisms through which gene expression noise arises. They constructed a model system in Saccharomyces cerevisiae to experimentally investigate the role of transcription scaffold stability and transcriptional reinitiation on the level of stochasticity in gene expression. They studied how these factors affect phenotypic heterogeneity both at the individual cell level as well as across a population of isogenic cells. They demonstrated experimentally that transcriptional bursting can have a significant effect on the level of stochasticity in eukaryotic gene expression. They also showed experimentally that transcription scaffold stability is linked to the level of stochasticity in the production of the downstream gene.
Mike Thompson, with D di Bernardo, Gardner, SE Chobot, EL Eastwood, AP Wojtovich, SJ Elliott, SE Schaus and Collins, developed and successfully demonstrated an algorithm that can identify the genes and proteins that mediate a cell's response to drug treatment. The algorithm enables the use of unstructured gene expression data to estimate a linear dynamic model of the gene regulatory network underlying drug response. The model is estimated using a recursive multiple regression scheme combined with singular value decomposition for de-noising and dimensional reduction. The estimated model is subsequently applied to new data measuring a cell's response to drug treatment. The model acts as a filter that separates the genes that are initial mediators of drug response from those genes that are only secondary responders to the drug. Using the method, they successfully predicted the molecular targets of multiple compounds, including a potential new anticancer compound, PTSB. PTSB inhibits growth in human small lung carcinoma cells and in the test organism (baker's yeast). In follow-up experiments, they verified that PTSB acts on thioredoxin and thioredoxin reductase, the molecular targets predicted by their algorithm. These findings have validated the algorithm's capabilities and facilitated investigation of a novel class of therapeutic compounds.
Jeremiah Faith, Simon Kasif and Gardner developed an algorithm for identifying transcriptional regulatory networks from whole-genome RNA expression data. Identifying the structure of regulatory pathways in bacteria has the potential to accelerate the development of novel antibiotics, and microbe-based biotechnologies. Mapping transcription factor regulatory networks is a critical first step toward this goal. Their novel algorithm is based on the principles of Bayesian networks. The approach is well suited to noisy, high-dimensional data sets, but may produce multiple solutions when the number of independent data points is limited. To address this limitation, they have enhanced the approach by adding a model averaging procedure. The algorithm first learns a group of probable models using a Markov Chain Monte Carlo search. It then averages a random sample of these models to determine the most likely regulatory interactions and confidence estimates for each. In addition, it employs a mixed linear/nonlinear model structure wherein the most appropriate regulatory relationship is automatically selected for each interaction. They successfully applied the new algorithm to a dataset of more than 300 Affymetrix E. coli microarrays obtained from more than 100 genetic or environmental perturbation experiments. They verified that the algorithm reconstructs major transcription factor/promoter interactions with high confidence. Moreover, they identified several previously undocumented targets of transcription factors, including new targets of the thoroughly studied LexA repressor.
Sensorimotor Dynamics
Attila Priplata, Jim Collins and colleagues from Afferent Corp. and Harvard Medical School investigated whether subsensory mechanical noise applied to the soles of the feet via vibrating insoles can be used to improve balance control in patients with diabetic neuropathy and patients with stroke. They also tested the hypothesis that baseline postural sway is directly correlated with both the sensory deficits of the individual and the amount of reduction of postural sway due to the application of input noise to the soles of the feet. Patients with diabetic neuropathy, patients with stroke and health elderly individuals were included in the study. It was found that application of noise resulted in a significant decrease in postural sway in the patients with diabetic neuropathy, patients with stroke, and the health individuals. It was also found that higher levels of baseline postural sway resulted in greater improvements in balance control with input noise.
Imaging of soft tissue
Mike S. Richards, working with Barbone, successfully defended his PhD prospectus entitled "Quantitative Three Dimensional Elasticity Imaging." Nachiket Gokhale, working jointly with AME faculty Assad Oberai and Barbone, successfully defended his PhD prospectus entitled "Nonlinear Elasticity Imaging." Richards and Nachiket worked together to produce the first three-dimensional quantitative reconstructions of elastic shear modulus from an ultrasound measured quasistatic deformation.
Ricardo Leiderman joined the Biomechanical Imaging Group as a postdoc in January. His background is in ultrasonic nondestructive evaluation, and he is training in biomechanics of soft tissues. He's currently coding mathematical models of soft tissue deformation, and will be supervising an REU summer student on calibrating nonlinear mechanical properties of different hydrogel recipes for use as tissue mimicking phantom materials.
Lindsey Nelson, under the supervision of Paul Barbone, last summer developed a suite of computer programs to extract deformation fields from image sequences.
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