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Research Interests and Publications
Application and development of reduction methods for neuroscience and biomechanics
After finishing my PhD at the University of Bristol, UK, I relocated to the Center for Biodynamics at Boston University in October 2001. I worked on neuro-related projects with Nancy Kopell, particularly the application of techniques from my PhD thesis to problems of synchronization in neural systems using Hodgkin-Huxley equations, and reductions thereof. Our aim was to gain a better mathematical understanding of how a high-dimensional system of equations can be accurately represented by a low-dimensional system. For instance, in [EK] a one-dimensional map captures the synchronization properties of a simplified model of hippocampus spiking at gamma frequencies (20-100Hz), and a goal is to implement an automatic method for generating and analyzing such maps.
I developed a MATLAB program called DSSRT that semi-automatically reduces a large class of coupled ODE systems to a sequence of lower-dimensional models, near an orbit of interest (such as a limit cycle computed numerically). This does not work by conventional PCA/SVD techniques. It does not assume explicit small parameters in the system. Instead, it tracks the relative "influence" of coupled variables on the dynamics of each variable, and then uses the relative time-scales of the variables (with knowledge of the differential equations) to determine the reduced regimes (see publications). It is therefore related to standard multiple-scale singular perturbation analysis, in the case of emerging scales of time and influence (arising from the state-dependent coupling). The technique has been validated so far with Hodgkin-Huxley neural models, but is well-suited to other systems of ODEs exhibiting non-expanding vector fields, particularly models of chemical kinetics.
The DSSRT program is in development, and stands for Dominant-Scale System Reduction Tool. Research papers explaining it in more detail have been published, and can be found here. The package can be downloaded here, suitable for use with Matlab R13 or higher. It includes five demo ODE systems (including a version of the FitzHugh-Nagumo oscillator) and documentation about the program's use and the theory of dominant scales it incorporates. The program includes tools for analysis of attractor basins and perturbed orbits, and some other useful features. See here for more information and full documentation, as well as inside the downloadable zip file.
In October 2004 I moved to Cornell University to continue this type of approach on hybrid systems involving neural control of locomotion, with John Guckenheimer. This work is funded by an NSF Frontiers in Integrative Biology grant. I have concentrated on data analysis techniques for estimating the dimension of dynamical systems from experimental data, for which we have a journal article under review (see below) and a poster from the 2006 World Congress on Biomechanics. The original motion capture data for the constrained hand tasks can be downloaded as an 8Mb .zip file here.
See also: Dimension Estimates for Attractors, John Guckenheimer, Contemporary Mathematics, Vol. 28, 1984.
Another project we are working on is the development of new software tools (based upon previous tools such as DsTool, XPP, AUTO) that can manipulate and analyze hybrid dynamical systems. This work is joint with Erik Sherwood, Drew LaMar, and John Guckenheimer at the Center for Applied Mathematics. The software is known as PyDSTool, and is being developed in Python. The software is in a beta release stage, and is available as "research code" to interested parties who are willing to help us test and develop it further. PyDSTool is hosted locally on a MoinMoin Wiki, and on SourceForge.
We are applying PyDSTool and other computational methods (e.g., from Matlab) to the study of locomotion in cockroaches (PolyPEDAL), and the biomechanics of hand movement (in the lab of Francisco Valero-Cuevas).
Other interesting references:
Numerical Computation in the Information Age by John Guckenheimer.
Abstract from our 2004 paper, "Epilepsy in Small-World Networks", by Theoden I. Netoff, Robert Clewley, Scott Arno, Tara Keck, John A. White, The Journal of Neuroscience, 24(37):8075-8083, 2004.
In Hippocampal slice models of epilepsy, two behaviors are seen: short bursts of electrical activity lasting 100 msec, and longer lasting electrical activity on the order of seconds resembling seizures. The bursts originate from the CA3 region, where there is a high degree of recurrent excitatory connections. Seizures originate from the CA1 where there are fewer recurrent connections (if the connections between the CA3 and CA1 are cut). To explain this behavior we have made model networks of excitatory neurons, using several types of model neurons. The model neurons were connected in a small-world network, where there are predominantly local connections and some long distance random connections. By changing parameters such as the synaptic strengths, number of synapses per neuron, proportion of local vs. long distance connections, we observed both epileptiform behaviors. Based on these simulations, we made a simple mathematical description of these networks, under some assumptions. This mathematical description explains why and when changes in the topology or synaptic strength cause transitions from "normal" to "seizing" to "bursting". These behaviors appear to be general properties of excitatory networks.
See the movies and the slides from a recent talk on this subject, which include detailed derivations for the one-dimensional and (1+R)-dimensional maps of the activity (not included in the paper), and references to recent papers on small-world network models of epilepsy.
While I was a teaching assistant for the 2004 Woods Hole "Methods in Computational Neuroscience" course I found that some of the students wanted to do sophisticated parameter sweeps and other tricks with their XPP simulations. In response, I wrote some basic Matlab functions that can change XPP .ode and .set files and run XPP externally to Matlab.
Keep backups of all your original .ode and .set files before letting any program (especially mine) stomp all over them!
The interface works only with versions of Matlab later than R13. If you want to try fixing the code for R12 or R11 yourself I expect the only thing you'll have to do is declare the following two variables at the beginning of every .m file provided in the package: true = 1; false = 0;
Please note that you may have to specify the full path to the XPP executable (in the 4th argument to the call to RunXPP) in order to get Matlab to run XPP. On Linux/OS X, depending on your installation, execute 'which xpp' or 'which xppaut'. On windows, look at the file properties for your shortcut to winPP or xppwin.
Note about output files: There is currently no facility to change integration parameters or anything that appears after the "@" command. This means that the data output will go to the same file after every external run. One way to get around this is to rename the output file after each run so that it won't be overwritten on the next run. This should cause little additional overhead. For instance:
root_name = 'out';
for i = 1:10
< run XPP here >
movefile('output.dat', [root_name num2str(i) '.dat'])
endVersion 070626 (26 June 2007):
* Added platform-specific line terminators to ChangeXPPodeFile.m and ChangeXPPsetFile.m
(\n for linux, unix, and OS X, and \r\n for windows)
* Updated error messages to be generated using the Matlab `error` command
* Added better check for termination of XPP (thanks to Michael Rempe for this)
* Changed the Mac OS X execution of xpp from using `!` command to `system`
Unfortunately some of these changes are untested so please alert me to any problems.
Also included in the download is a function to generate an array of spike times (or more generally, arbitrary threshold crossings) from simulation data, and an example script showing how these functions can be used to generate a 3D plot of a neuron's firing frequency response from varying two parameters simultaneouly. Documentation is included in the download, and can also be read online here.
I have found these functions very useful for a variety of computational problems, although they are no feat of programming or imagination on my part. Essentially, these functions let you call XPP in "silent mode" from within Matlab, and let you change parameter values or initial conditions within your .ode or .set files. Running XPP in silent mode means that it dumps its output to an ASCII-format file which can be easily loaded back into Matlab for analysis.
This may sound like a long-winded and slow process, but XPP typically integrates your models faster than the native Matlab DE solvers, and the models are more intuitively specified using XPP anyway (in my opinion). I have found there to be minimal overhead involved in having my functions change parameters by automatically editing the .ode or .set files in situ, or loading modestly sized data files from a hard drive.
This interface can be useful to Matlab users for a variety of reasons:
(a) If you are not an advanced XPP user, and you don't understand how to set up multi-parameter / initial condition (I.C.) range integration batch jobs in XPP, or you find doing so too fussy and prone to mistakes. You may want to vary more than 2 parameters or I.C.s. without specifying a large table of pre-determined values within XPP, or you want to allow a user to set up the ranges interactively in Matlab.
(b) Furthermore, you might want to run Matlab scripts that adaptively select parameter values or I.C.s based on the results of previous XPP integration results (e.g. parameter estimation algorithms, "shooting" methods, etc.).
(c) You can do data analysis on the integrated orbits without writing dynamic link libraries (usually in C) that are called from within your .ode script. (Some institutional installations of XPP may not have been done with the DLL option set, which can be a problem.) Also, you can take advantage of Matlab's built-in statistical tools and visualization capabilities.
(d) You don't need an X server on a Windows platform to run XPP in silent mode, so I guess these functions might make XPP a little more usable in the event that you don't have a working X server in Windows!
A major goal of modern theoretical science is being able to describe and explain processes of natural self-organisation in physics, biology or psychology. If only from a pragmatic view, such an understanding could allow us to design our own "naturally" self-organising systems, for a more integrated engineering approach to real world problems. This would include the development of more "naturally"-inspired intelligent systems, that solve the problems of robustness, flexibility, and computation in a similar way to the brain.
In more detail:
Also see Kelby Mason's great paper on Daniel Dennett and memetics. Also see the Journal of Memetics UK website, the UK Memes website (especially the Meme Lab page), and the Principia Cybernetica Project website.
Despite the achievements of modern science in the extreme scales of quantum and relativistic physics, a profound issue remains unresolved in the domain of normal human experience. In contrast to the well-known counter-intuitive picture of the universe that quantum and relativistic physics paints, it is easy to forget that the science of the objects and interactions of normal human experience is not unified in its foundations either. There can be significant incompatibility between scientific theories in this realm which have overlapping domains but which use language based on different representations of natural features.
For example, the fact that the mind-body problem remain unresolved means that theories of human behaviour and neural behaviour must use very different elementary objects and interactions. Explaining the diversity of organic life based on an analysis of pre-biotic chemistry has similar difficulties. It is appealing to consider the possibility of an "ontological unity" [Kan] that might, for instance, lead to a theory having one set of basic representations which could ultimately underlie all accounts of the observations at the levels of both neurobiology and social behaviour. The exact meaning of the term "ontological unity" can be found in [Cle]. The reasons why the issue of "scientific unity" (in general) is still considered a dilemma are addressed in the link. There, it is also mentioned how those reasons motivate the form of a scientific methodology which will be central to the achievement of the goals outlined above.
[ See here for continuation ]
My PhD was obtained in the Engineering Mathematics department of Bristol University, in the UK. My supervisor was Prof. Alan Champneys. The title of the thesis is "Techniques for analysing multistability in spiking neural networks", and can be downloaded in PDF and Gzipped PostScript format (US letter format).
Here is the thesis abstract:
In this thesis we consider techniques to understand transient and global dynamics in networks of spiking neurons. We focus on a form of multistability inspired from observations in experiments, and analyse its existence, stability and control using simulations, numerical methods and rigorous analysis of a model network. This network is designed to exemplify certain properties found in real networks of the brain, and is based on a dynamical systems approach to mathematical modelling, in which neural networks are represented by coupled systems of ordinary differential equations. We introduce the background to making an appropriate choice of model for studying multistable dynamics in the brain, and fix a model for use throughout the thesis. This model is based on an extension of the `integrate-and-fire' reduced neuron, and the idea of `locally coupled' networks of many neurons (e.g. between five and one hundred) as a simplified view of modular networks found in parts in the brain.
In the networks we consider we retain biological features that have recently become the subject of great interest, in terms of their role in the possible mechanisms of neural processing in the brain. Throughout the thesis, the effects that these additional features have on the dynamics we can observe are compared with corresponding models that do not include these features. The inclusion of these features presents interesting computational and analytical challenges. We discuss an approach to analysing systems of this nature through a set of `inter-locking' local analyses, which can be used to provide inter-related sufficient conditions that prescribe global properties of our network. Many aspects of this analysis are approximate in nature, but their validity is subject to explicit bounds that can be calculated, and nevertheless yields a powerful insight into the global properties of multistability in the network.
The results in this thesis have relevance both beyond the particular network considered, and beyond computational neuroscience in general. Firstly, we discuss experimentally-driven research in neuroscience that may benefit from the modelling techniques developed here. Secondly, as a study of a nonlinear system of coupled oscillators, this thesis develops ideas that may have application in studying complex pattern formation and self-organizing phenomena in other disciplines.
This work extends studies of synchronisation and oscillator suppression/death for biological neural networks in two ways. Firstly, it is shown that robustness in the mechanism is possible only when realistic conductance-based synaptic inputs and absolute refractoriness are included in the model. Secondly, analysis of both the transient and long term steady state behaviour of the network remains possible even with these added features. The analysis accurately predicts parameter regimes within which the network works robustly for a wide range of inputs. The work is being extended in application to carefully reduced models of hippocampal synchronization, using Hodgkin-Huxley type equations.
Paper in the Proceedings of the 15th International Congress on Cybernetics (Association Internationale de Cybernetique, Namur)...
Emergence Without Magic: The Role of Memetics in Multi-Scale Models of Evolution and Behaviour (with minor corrections -- August 1998)
A detailed version of this paper, including a broader analysis of memetic theories, is in preparation as a two part essay. A version is available in html format: Conceputal Problems in Memetics in a Multi-Level View of Evolution and Behaviour and Part II. The papers introduce some concepts from scientific methodology which are discussed in greater detail here.