Transient response in a somatic shunt cable model for synaptic input activated at the terminal

Regulation of Mitoflash Biogenesis and Signaling by Mitochondrial Dynamics

Mitochondria are highly dynamic organelles undergoing constant network reorganization and exhibiting stochastic signaling events in the form of mitochondrial flashes (mitoflashes). Here we investigate whether and how mitochondrial network dynamics regulate mitoflash biogenesis and signaling. We found that mitoflash frequency was largely invariant when network fragmentized or redistributed in the absence of mitofusin (Mfn) 1, Mfn2, or Kif5b. However, Opa1 deficiency decreased spontaneous mitoflash frequency due to superimposing changes in respiratory function, whereas mitoflash response to non-metabolic stimulation was unchanged despite network fragmentation. In Drp1- or Mff-deficient cells whose mitochondria hyperfused into a single whole-cell reticulum, the frequency of mitoflashes of regular amplitude and duration was again unaltered, although brief and low-amplitude “miniflashes” emerged because of improved detection ability. As the network reorganized, however, the signal mass of mitoflash signaling was dynamically regulated in accordance with the degree of network connectivity. These findings demonstrate a novel functional role of mitochondrial network dynamics and uncover a magnitude- rather than frequency-modulatory mechanism in the regulation of mitoflash signaling. In addition, our data support a stochastic trigger model for the ignition of mitoflashes.

Neuronal integrative analysis of the "Dumbbell" model for passive neurons

We analyze the so called "Dumbbell" model of Jackson (J. Neurophysiol. 69 (1993) pp. 464) for a single neuron consisting of a patch-clamped cell body attached to dendritic cable of finite length terminating in an oblique derivative ("natural termination") boundary condition representing a dendritic swelling or a natural ending sealed by a continuous surface of the cell membrane. The model is solved analytically via the Green's function method. Large and small time asymptotic behavior of the membrane potential is developed when there is a somatic voltage-clamp imposed. We discuss the difference in the voltage distribution if a sealed-end (Neumann) termination is used instead of the natural termination boundary condition. If the access resistance is large the differences between the potentials corresponding to the two boundary conditions are small at the soma, but can vary significantly near the dendritic termination. This discrepancy is amplified at the soma if there is a synaptic stimulus introduced between the soma and dendritic tip.

Contribution of voltage-dependent potassium channels to the somatic shunt in neck motoneurons of the cat

The specific membrane resistivity of motoneurons at or near the soma (Rms) is much lower than the specific membrane resistivity of the dendritic tree (Rmd). The goal of the present experiments was to investigate the contribution of tonically active voltage-dependent potassium channels at or near the soma to the low Rms. These channels were blocked with the use of intracellular injections of cesium. Input resistance (RN), Rms/Rmd, a conductance representing voltage-dependent potassium channels on the soma, GK, and a conductance attributed to damage caused by electrode impalement, GDa, were estimated from voltage responses to a step of current. The effect of intracellular injections of cesium on electrotonic structure was determined with the use of two strategies: 1) a population analysis that compared data from two groups of motoneurons, those recorded with potassium acetate electrodes (control group) and those recorded with cesium acetate electrodes after the motoneurons were loaded with cesium; and 2) an analysis of changes in electrotonic structure that occurred over the course of multiple injections of cesium or during similar periods of time in control cells. RN of control and cesium-loaded motoneurons was similar. Over the course of the recordings, RN of control cells usually increased and this increase was associated with a hyperpolarization. In contrast, increases in RN after successive injections of cesium were closely linked to a depolarization. At corresponding membrane potentials, Rms/Rmd of cesium-loaded motoneurons was greater than Rms/Rmd of control motoneurons. Over the course of cesium injections, Rms/Rmd increased and the membrane potential depolarized. In contrast, increases in Rms/Rmd observed during the course of recordings from control cells were associated with a hyperpolarization. Compared with control cells at corresponding membrane potentials, GK was less in cesium-loaded cells. Increases in GK that occurred over the course of recordings in control cells were closely coupled to a depolarization. In cesium-loaded cells, GK usually decreased as the cell depolarized during the injections. In control cells, increases in GDa that occurred during the recording period were closely coupled to a depolarization. In contrast, changes in GDa that occurred during cesium injections were not related to the change in membrane potential and did not explain the corresponding changes in Rms/Rmd and membrane potential. The results of this study indicate that voltage-dependent potassium channels contribute to the somatic shunt (low Rms) in neck motoneurons and provide a voltage-dependent mechanism for the dynamic regulation of motoneuron electrotonic properties.

Techniques for the application of the analytical solution to the multicylinder somatic shunt cable model for passive neurones

The general solution for the voltage response to a generic impulse current input in a multicylinder somatic shunt cable model for passive neurones has been developed in [1]. In this paper we consider the application of the multicylinder solution to examples previously considered by other authors for the single cylinder case: long and short current input and synaptic input modeled by an alpha-function and a multi-exponential function. Simple expansions appropriate for small and large times are found and efficient means of obtaining these expansions are clearly demonstrated. The dependence of the small and large time solutions upon the dimensionless parameters appearing in the conservation of current condition at the soma is investigated. Relevant limits of these dimensionless parameters which further simplify the small and large time solutions are related back to equivalent dimensional problems of interest to the practitioner. The well-posedness of the dimensionless inverse problem is investigated and a method proposed for the solution of the dimensional inverse problem for the somatic shunt.

Techniques for obtaining analytical solutions to the multicylinder somatic shunt cable model for passive neurones

The somatic shunt cable model for neurones is extended to the case in which several equivalent cylinders, not necessarily of the same electrotonic length, emanate from the cell soma. The cable equation is assumed to hold in each cylinder and is solved with sealed end conditions and a lumped soma boundary condition at a common origin. A Green's function (G) is defined, corresponding to the voltage response to an instantaneous current pulse at an arbitrary point along one of the cylinders. An eigenfunction expansion for G is obtained where the coefficients are determined using the calculus of residues and compared with an alternative method of derivation using a modified orthogonality condition. This expansion converges quickly for large time, but, for small time, a more convenient alternative expansion is obtained by Laplace transforms. The voltage response to arbitrary currents injected at arbitrary sites in the dendritic tree (including the soma) may then be expressed as a convolution integral involving G. Illustrative examples are presented for a point charge input.

A non-uniform equivalent cable model of membrane voltage changes in a passive dendritic tree

A non-uniform equivalent cable model of membrane voltage changes in a passive dendritic tree extending Rall's equivalent cylinder model is presented. It is obtained from a combination of cable theory with the continuum approach. Replacing the fine structure of the branching dendrites by an equivalent, conductive medium characterized by averaged electrical parameters, the one-dimensional cable equations with spatially varying parameters are derived. While these equations can be solved in general only numerically, we were able to formulate a general branching condition (comprising Rall's 3/2 power relationship as a special case) under which analytical solutions can be deduced from those of the equivalent cylinder model. This model allows dendritic trees with a greater variety of branching patterns than before to be analytically treated.