Model reduction of strong-weak neurons.

We consider neurons with large dendritic trees that are weakly excitable in the sense that back propagating action potentials are severly attenuated as they travel from the small, strongly excitable, spike initiation zone. In previous work we have shown that the computational size of weakly excitable cell models may be reduced by two or more orders of magnitude, and that the size of strongly excitable models may be reduced by at least one order of magnitude, without sacrificing the spatio-temporal nature of its inputs (in the sense we reproduce the cell's precise mapping of inputs to outputs). We combine the best of these two strategies via a predictor-corrector decomposition scheme and achieve a drastically reduced highly accurate model of a caricature of the neuron responsible for collision detection in the locust.

Structure-preserving model reduction of passive and quasi-active neurons.

The spatial component of input signals often carries information crucial to a neuron's function, but models mapping synaptic inputs to the transmembrane potential can be computationally expensive. Existing reduced models of the neuron either merge compartments, thereby sacrificing the spatial specificity of inputs, or apply model reduction techniques that sacrifice the underlying electrophysiology of the model. We use Krylov subspace projection methods to construct reduced models of passive and quasi-active neurons that preserve both the spatial specificity of inputs and the electrophysiological interpretation as an RC and RLC circuit, respectively. Each reduced model accurately computes the potential at the spike initiation zone (SIZ) given a much smaller dimension and simulation time, as we show numerically and theoretically. The structure is preserved through the similarity in the circuit representations, for which we provide circuit diagrams and mathematical expressions for the circuit elements. Furthermore, the transformation from the full to the reduced system is straightforward and depends on intrinsic properties of the dendrite. As each reduced model is accurate and has a clear electrophysiological interpretation, the reduced models can be used not only to simulate morphologically accurate neurons but also to examine computations performed in dendrites.

Inverse problems in neuronal calcium signaling.

Calcium is the most important of the brain's second messengers. Thanks to engineered fluorescent indicators and caged compounds we have an excellent qualitative picture of its regulation and impact.With the advent of new scanning technology that permits one to observe the calcium signal throughout a highly branched neuron the potential exists for functional, single cell, quantitative calcium imaging. To help realize that potential we analyze a sequence of four inverse problems that infer the parameters of the cytosolic calcium buffers and plasma membrane calcium pumps and channels from the light shed by fluorescent indicators following specific stimulus protocols. Our analyses lead in each case to practical algorithms that we illustrate and test on synthetic data.

Morphologically accurate reduced order modeling of spiking neurons.

Accurately simulating neurons with realistic morphological structure and synaptic inputs requires the solution of large systems of nonlinear ordinary differential equations. We apply model reduction techniques to recover the complete nonlinear voltage dynamics of a neuron using a system of much lower dimension. Using a proper orthogonal decomposition, we build a reduced-order system from salient snapshots of the full system output, thus reducing the number of state variables. A discrete empirical interpolation method is then used to reduce the complexity of the nonlinear term to be proportional to the number of reduced variables. Together these two techniques allow for up to two orders of magnitude dimension reduction without sacrificing the spatially-distributed input structure, with an associated order of magnitude speed-up in simulation time. We demonstrate that both nonlinear spiking behavior and subthreshold response of realistic cells are accurately captured by these low-dimensional models.

Low-dimensional, morphologically accurate models of subthreshold membrane potential.

The accurate simulation of a neuron's ability to integrate distributed synaptic input typically requires the simultaneous solution of tens of thousands of ordinary differential equations. For, in order to understand how a cell distinguishes between input patterns we apparently need a model that is biophysically accurate down to the space scale of a single spine, i.e., 1 mum. We argue here that one can retain this highly detailed input structure while dramatically reducing the overall system dimension if one is content to accurately reproduce the associated membrane potential at a small number of places, e.g., at the site of action potential initiation, under subthreshold stimulation. The latter hypothesis permits us to approximate the active cell model with an associated quasi-active model, which in turn we reduce by both time-domain (Balanced Truncation) and frequency-domain (H(2) approximation of the transfer function) methods. We apply and contrast these methods on a suite of typical cells, achieving up to four orders of magnitude in dimension reduction and an associated speed-up in the simulation of dendritic democratization and resonance. We also append a threshold mechanism and indicate that this reduction has the potential to deliver an accurate quasi-integrate and fire model.

Direct correction of non-space-clamped currents via Cole's theorem.

The currents measured during voltage clamp of a non-isopotential neuron reflect axial as well as membrane conductances. One wishes to remove the former in the hope that the latter will reveal quantitative information on the nature of the voltage gated channels at the clamp site. We here show that Cole's theorem can be used to directly remove the axial component from simulated voltage clamp recordings. It is direct in the sense that it requires neither simulation nor fitting. As it comes down to squaring the clamp current and then differentiating with respect to clamp voltage it may indeed be implemented in real time. When applied to synthetic potassium currents in straight fibers we find that our method accurately and robustly recovers both non-uniform conductances and non-uniform channel kinetics. We indicate the degree to which such accuracy is diminished for cells that taper and/or branch.

A kinetic model of oxygen regulation of cytochrome production in Escherichia coli.

Recent experimental work has identified the principal components arrayed by Escherichia coli in its sensing of, and response to, varying levels of oxygen. This apparatus may be leveraged/modified by the metabolic engineer to identify nonuniform oxygen and glucose regimens that deliver better yields than their uniform counterparts. Toward this end we build and analyse a mathematical model that captures the role played by oxygen in the regulation of cytochrome production in E. coli.

An adjoint method for channel localization.

Single cells learn by tuning their synaptic conductances and redistributing their excitable machinery. To reveal its learning rules, one must therefore know how the cell remaps its ion channels in response to physiological stimuli. We here develop an adjoint approach for discerning the non-uniform distribution of a given channel type from knowledge of the time course of membrane potential at two distinct locations following a prescribed injection of current.

Development of a metabolic network design and optimization framework incorporating implementation constraints: a succinate production case study.

We have developed a pathway design and optimization scheme that accommodates genetically and/or environmentally derived operational constraints. We express the full set of theoretically optimal pathways in terms of the underlying elementary flux modes and then examine the sensitivity of the optimal yield to a wide class of physiological perturbations. Though the scheme is general it is best appreciated in a concrete context: we here take succinate production as our model system. The scheme produces novel pathway designs and leads to the construction of optimal succinate production pathway networks. The model predictions compare very favorably with experimental observations.

Genetically constrained metabolic flux analysis.

Significant progress has been made in using existing metabolic databases to estimate metabolic fluxes. Traditional metabolic flux analysis generally starts with a predetermined metabolic network. This approach has been employed successfully to analyze the behaviors of recombinant strains by manually adding or removing the corresponding pathway(s) in the metabolic map. The current work focuses on the development of a new framework that utilizes genomic and metabolic databases, including available genetic/regulatory network structures and gene chip expression data, to constrain metabolic flux analysis. The genetic network consisting of the sensing/regulatory circuits will activate or deactivate a specific set of genes in response to external stimulus. The activation and/or repression of this set of genes will result in different gene expression levels that will in turn change the structure of the metabolic map. Hence, the metabolic map will automatically "adapt" to the external stimulus as captured by the genetic network. This adaptation selects a subnetwork from the pool of feasible reactions and so performs what we term "environmentally driven dimensional reduction." The Escherichia coli oxygen and redox sensing/regulatory system, which controls the metabolic patterns connected to glycolysis and the TCA cycle, was used as a model system to illustrate the proposed approach.

Estimating the location and time course of synaptic input from multi-site potential recordings.

A method is introduced that permits accurate and robust extraction of the location and time course of synaptic conductance from potentials recorded on either side of, and perhaps at some distance from, the synapse in question. It is shown that such data permits one to fully overcome the problems typically associated with lack of spaceclamp. The method does not presume anything about the nature of the time course and yet is applicable to branched, active cells receiving simultaneous input from a number of synapses.

Recovering the passive properties of tapered dendrites from single and dual potential recordings.

We demonstrate that measurement of the membrane potential at one or more sites on a branched and tapered neuron following a known transient injection of subthreshold somatic current uniquely determines the cell's passive electrical properties. That is, knowledge of the potentials allows recovery of the cell's axial resistance, membrane capacitance, membrane conductance and soma conductance. The argument underlying uniqueness leads immediately to a constructive, robust algorithm that we successfully test on synthetic data. The robustness stems from the fact that the algorithm requires only a few weighted integrals, or moments, of the measured potentials.

Estimating the time course of pore expansion during the spike phase of exocytotic release in mast cells of the beige mouse.

Our objective is to determine the time course of exocytotic fusion pore opening (P) in mast cells of the beige mouse from the measured efflux of the spike phase of exocytotic release (J). We show that a pore whose meridian or radius grows linearly with time cannot reproduce the efflux. We also show that a pore that opens very quickly [relative to the diffusivity of 5-hydroxytryptamine (5-HT)] and completely (P = pi) also does not mimic the experimental efflux, and estimate maximum pore angles of 70 (+/- 20) degrees. We show that a larger class of opening functions reproduces the rising phase and part of the decay phase and calculate pore expansion rate, pore radius and pore angle, none of which can be readily measured. In the initial stages of the spike phase (50-200 ms) when the gel matrix has not expanded significantly, this model suggests that the pore radius increases exponentially with a time constant of 82(+/- 62) ms with pore expansion reaching its maximum velocity of 20 (+/- 7) nm ms-1. We conclude that the release process is dynamic and suggest that the velocity of pore opening (V) and the diffusivity of 5-HT (D), in addition to the size of the vesicle (R, radius), vary with time. We discuss assumptions and improvements to the model and propose that this methodology is applicable for determining P from measured J in other endocrine cells and neurons when D within the secretory vesicle is much less than D within the pore neck.