Parameter estimation methods for single neuron models

With the advancement in computer technology, it has become possible to fit complex models to neuronal data. In this work, we test how two methods can estimate parameters of simple neuron models (passive soma) to more complex ones (neuron with one dendritic cylinder and two active conductances). The first method uses classical voltage traces resulting from current pulses injection (time domain), while the second uses measures of the neuron's response to sinusoidal stimuli (frequency domain). Both methods estimate correctly the parameters in all cases studied. However, the time-domain method is slower and more prone to estimation errors in the cable parameters than the frequency-domain method. Because with noisy data the goodness of fit does not distinguish between different solutions, we suggest that running the estimation procedure a large number of times might help find a good solution and can provide information about the interactions between parameters. Also, because the formulation used for the model's response in the frequency domain is analytical, one can derive a local sensitivity analysis for each parameter. This analysis indicates how well a parameter is likely to be estimated and helps choose an optimal stimulation protocol. Finally, the tests suggest a strategy for fitting single-cell models using the two methods examined.

A model of the action potential and underlying membrane currents in a rabbit atrial cell

We have developed a mathematical model of the rabbit atrial myocyte and have used it in an examination of the ionic basis of the atrial action potential. Available biophysical data have been incorporated into the model to quantify the specific ultrastructural morphology, intracellular ion buffering, and time- and voltage-dependent currents and transport mechanisms of the rabbit atrial cell. When possible, mathematical expressions describing ionic currents identified in rabbit atrium are based on whole cell voltage-clamp data from enzymatically isolated rabbit atrial myocytes. This membrane model is coupled to equations describing Na+, K+, and Ca2+ homeostasis, including the uptake and release of Ca2+ by the sarcoplasmic reticulum and Ca2+ buffering. The resulting formulation can accurately simulate the whole cell voltage-clamp data on which it is based and provides fits to a family of rabbit atrial cell action potentials obtained at 35 degrees C over a range of stimulus rates (0.2-3.0 Hz). The model is utilized to provide a qualitative prediction of the intracellular Ca2+ concentration transient during the action potential and to illustrate the interactions between membrane currents that underlie repolarization in the rabbit atrial myocyte.

Quantitative analysis of electrotonic structure and membrane properties of NMDA-activated lamprey spinal neurons

Parameter optimization methods were used to quantitatively analyze frequency-domain-voltage-clamp data of NMDA-activated lamprey spinal neurons simultaneously over a wide range of membrane potentials. A neuronal cable model was used to explicitly take into account receptors located on the dendritic trees. The driving point membrane admittance was measured from the cell soma in response to a Fourier synthesized point voltage clamp stimulus. The data were fitted to an equivalent cable model consisting of a single lumped soma compartment coupled resistively to a series of equal dendritic compartments. The model contains voltage-dependent NMDA sensitive (INMDA), slow potassium (IK), and leakage (IL) currents. Both the passive cable properties and the voltage dependence of ion channel kinetics were estimated, including the electronic structure of the cell, the steady-state gating characteristics, and the time constants for particular voltage- and time-dependent ionic conductances. An alternate kinetic formulation was developed that consisted of steady-state values for the gating parameters and their time constants at half-activation values as well as slopes of these parameters at half-activation. This procedure allowed independent restrictions on the magnitude and slope of both the steady-state gating variable and its associated time constant. Quantitative estimates of the voltage-dependent membrane ion conductances and their kinetic parameters were used to solve the nonlinear equations describing dynamic responses. The model accurately predicts current clamp responses and is consistent with experimentally measured TTX-resistant NMDA-induced patterned activity. In summary, an analysis method is developed that provides a pragmatic approach to quantitatively describe a nonlinear neuronal system.

Localization and interaction of N-methyl-D-aspartate and non-N-methyl-D-aspartate receptors of lamprey spinal neurons

Small volumes of N-Methyl-D-Aspartate (NMDA) and non-NMDA excitatory amino acid receptor agonists were applied to localized regions of the dendritic trees of lamprey spinal neurons along their medial-lateral axis to obtain a spatial map of glutamate receptor distribution. Voltage clamp and frequency domain methods were used to obtain quantitative kinetic data of the voltage dependent ionic channels located both on the soma and on highly branched dendritic membranes. Pressure pulses of NMDA applied to the most peripheral regions of the dendritic tree elicited large somatic impedance increases, indicating that the most peripheral dendrites are well supplied with NMDA receptors. Experiments done with kainate did not elicit somatic responses to agonist applications on peripheral dendrites. The data obtained are consistent with the hypothesis that the activation of NMDA receptors by exogenous glutamate is significantly modified by the simultaneous activation of non-NMDA receptors, which shunts the NMDA response. The non-NMDA shunting hypothesis was tested by a combined application of kainate and NMDA to mimic the action of glutamate showing that the shunting effect of non-NMDA receptor activation virtually abolished the marked voltage dependency typical of NMDA receptor activation. These data were interpreted with a compartmental neuronal model having both NMDA and non-NMDA receptors.

A mathematical model of a rabbit sinoatrial node cell

A mathematical model for the electrophysiological responses of a rabbit sinoatrial node cell that is based on whole cell recordings from enzymatically isolated single pacemaker cells at 37 degrees C has been developed. The ion channels, Na(+)-K+ and Ca2+ pumps, and Na(+)-Ca2+ exchanger in the surface membrane (sarcolemma) are described using equations for these known currents in mammalian pacemaker cells. The extracellular environment is treated as a diffusion-limited space, and the myoplasm contains Ca(2+)-binding proteins (calmodulin and troponin). Original features of this model include 1) new equations for the hyperpolarization-activated inward current, 2) assessment of the role of the transient-type Ca2+ current during pacemaker depolarization, 3) inclusion of an Na+ current based on recent experimental data, and 4) demonstration of the possible influence of pump and exchanger currents and background currents on the pacemaker rate. This model provides acceptable fits to voltage-clamp and action potential data and can be used to seek biophysically based explanations of the electrophysiological activity in the rabbit sinoatrial node cell.

Conduction in bullfrog atrial strands: simulations of the role of disc and extracellular resistance

A number of fundamental properties of intercellular conduction in simulated cylindrical strands of cardiac tissue are examined. The paper is based on recent biophysical information describing the transmembrane ionic currents in bullfrog atrial cells as well as anatomical data on the structures (gap junctions) responsible for the coupling between cells in that tissue. A mathematical model of the single bullfrog atrial cell based on suction microelectrode single-cell voltage clamp data is employed, as well as a modified version of the well-known model of Heppner and Plonsey, to characterized the resistive connections between adjacent cells in a cardiac strand. In addition, the simulated cellular strand is assumed to be encased in a cylindrical, resistive endothelial sheath, thus forming an idealized atrial trabeculum; the trabeculum is immersed in an extensive volume conductor. It is possible to simulate both uniform and discontinuous conduction in this atrial strand model by appropriately changing the resistance of the intercalated discs that occur at cell boundaries. The conduction velocity achieved in the normal or control case is within the range of conduction velocities that have been measured for bullfrog atrial trabeculae using optical methods. Extracellular resistance is shown to have a significant effect on both conduction velocity and the critical value of disc resistance at which discontinuous conduction first occurs. Since the atrial cell model employed in this study is based on experimental data and can accurately simulate the atrial action potential, the transmembrane ionic currents generated by the model are capable of providing detailed information concerning the mechanisms of intercellular current spread, particularly in the region of the intercalated disc.

A model of slow conduction in bullfrog atrial trabeculae

The success or failure of the propagation of electrical activity in cardiac tissue is dependent on both cellular membrane characteristics and intercellular coupling properties. This paper considers a linear arrangement of individual bullfrog atrial cells that are resistively coupled end to end to form a cylindrical strand. The strand, in turn, is encased by an endothelial sheath that provides a restricted extracellular space and an ion diffusion barrier to the outer bathing medium. This encased strand serves as an idealized model of an atrial trabeculum. Excitable membrane characteristics of the atrial cell are specified in terms of a Hodgkin-Huxley type of model that is quantitatively based on single-microelectrode voltage clamp data from bullfrog atrial myocytes. This membrane model can simulate the behavior of normal cells as well as of ischemic cells that exhibit depressed electrophysiological behavior (e.g., decreased resting potential, upstroke velocity, peak height, and action potential duration). Depressed activity can be easily simulated with variation of a single model parameter, the gain of the Na+/K+ pump current (INaK). Intercellular coupling properties are specified in terms of a lumped resistive T-type network between adjacent cells. The atrial strand model provides a means for studying the theoretical aspects of slow conduction in a "hybrid" strand that consists of a central region of cells having abnormal membrane or coupling properties, flanked on either side by normal atrial cells. Both uniform and discontinuous conduction are simulated by means of appropriate changes in the coupling resistance between cells. In addition, by varying either the degree of depressed electrical activity or the intercalated disc resistance in the central zone of the strand, slow conduction or complete conduction block in that region is demonstrated. Since the cellular model used in this study is based on experimental data and closely mimics both the atrial action potential and the underlying membrane currents, it has the potential to (1) accurately represent the current and voltage wave-forms occurring in the region of intercalated discs and (2) provide detailed information regarding the mechanisms in intercellular current spread in the region of slow conduction.