An adjoint method for channel localization

Cox, Steven J.

doi:10.1093/imammb/dql004

Cited by 9 publications

(5 citation statements)

References 25 publications

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“…The third and the fourth optimality conditions form the adjoint equations for the constraints. We refer readers to Cox (2006) for more details about the adjoint method. These adjoint equations and their side conditions are…”

Section: First-order Optimality Conditionsmentioning

confidence: 99%

Reconstructing parameters of the FitzHugh–Nagumo system from boundary potential measurements

Keyes

2007

J Comput Neurosci

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We consider distributed parameter identification problems for the FitzHugh-Nagumo model of electrocardiology. The model describes the evolution of electrical potentials in heart tissues. The mathematical problem is to reconstruct physical parameters of the system through partial knowledge of its solutions on the boundary of the domain. We present a parallel algorithm of Newton-Krylov type that combines Newton's method for numerical optimization with Krylov subspace solvers for the resulting Karush-Kuhn-Tucker system. We show by numerical simulations that parameter reconstruction can be performed from measurements taken on the boundary of the domain only. We discuss the effects of various model parameters on the quality of reconstructions.

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Section: First-order Optimality Conditionsmentioning

confidence: 99%

Reconstructing parameters of the FitzHugh–Nagumo system from boundary potential measurements

Keyes

2007

J Comput Neurosci

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“…These methods often lack capabilities in live animals, and the dynamic relationship between the observed optical response and underlying neuronal activity is difficult to understand or quantify, given the introduction of foreign molecules into the cells. Nevertheless, their advent prompted advances in inverse problems that led to new algorithms for recovering distributions of ion channels, calcium concentrations and other intra-cellular molecular concentrations from these data ( Cox, 2006 ; Burger, 2011 ; Raol & Cox, 2013 ).…”

Section: Big Neuroscience and New Mathematicsmentioning

confidence: 99%

Will big data yield new mathematics? An evolving synergy with neuroscience

Feng

Holmes

2016

IMA J Appl Math

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New mathematics has often been inspired by new insights into the natural world. Here we describe some ongoing and possible future interactions among the massive data sets being collected in neuroscience, methods for their analysis and mathematical models of the underlying, still largely uncharted neural substrates that generate these data. We start by recalling events that occurred in turbulence modelling when substantial space-time velocity field measurements and numerical simulations allowed a new perspective on the governing equations of fluid mechanics. While no analogous global mathematical model of neural processes exists, we argue that big data may enable validation or at least rejection of models at cellular to brain area scales and may illuminate connections among models. We give examples of such models and survey some relatively new experimental technologies, including optogenetics and functional imaging, that can report neural activity in live animals performing complex tasks. The search for analytical techniques for these data is already yielding new mathematics, and we believe their multi-scale nature may help relate well-established models, such as the Hodgkin–Huxley equations for single neurons, to more abstract models of neural circuits, brain areas and larger networks within the brain. In brief, we envisage a closer liaison, if not a marriage, between neuroscience and mathematics.

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“…In [4,39,9,1,2], the question of determining spatially distributed conductances is investigated through different techniques and algorithms. They differ considerably from our method, and seem harder to generalize for other situations, as, for instance, when the domain is given by trees (with the obvious exception of [1,2]), for time dependent conductances, and for general nonlinear equations, our ultimate goal.…”

Section: Introductionmentioning

confidence: 99%

A computational approach for the inverse problem of neuronal conductances determination

2020

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The derivation by Alan Hodgkin and Andrew Huxley of their famous neuronal conductance model relied on experimental data gathered using neurons of the giant squid.It becomes clear that determining experimentally the conductances of neurons is hard, in particular under the presence of spatial and temporal heterogeneities. Moreover it is reasonable to expect variations between species or even between types of neurons of a same species. Determining conductances from one type of neuron is no guarantee that it works across the board.We tackle the inverse problem of determining, given voltage data, conductances with nonuniform distribution computationally. In the simpler setting of a cable equation, we consider the Landweber iteration, a computational technique used to identify non-uniform spatial and temporal ionic distributions, both in a single branch or in a tree. Here, we propose and (numerically) investigate an iterative scheme that consists in numerically solving two partial differential equations in each step. We provide several numerical results showing that the method is able to capture the correct conductances given information on the voltages, even for noisy data.

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An adjoint method for channel localization

Cited by 9 publications

References 25 publications

Reconstructing parameters of the FitzHugh–Nagumo system from boundary potential measurements

Reconstructing parameters of the FitzHugh–Nagumo system from boundary potential measurements

Will big data yield new mathematics? An evolving synergy with neuroscience

A computational approach for the inverse problem of neuronal conductances determination

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