Jürgen F. Fohlmeister

Modeling the repetitive firing of retinal ganglion cells

A kinetic model for the repetitive firing of retinal ganglion cells was synthesized from voltage-clamp data and evaluated by comparison with whole cell recordings from ganglion cells in the intact tiger salamander retina. Five distinct channels were included in the model and were sufficient to describe the physiologically observed frequency/current relationship in response to various levels of cell depolarization.

A Model for Phasic and Tonic Repetitively Firing Neuronal Encoders

A model for neuronal encoders is presented. Its mathematical description consists of two coupled first order, non-linear differential equations giving the time course of the membrane potential and of a leak function under conditions of continuous drive. The leak function is given by the ratio of the transmembrane conductance to the effective encoder capacitance and is closely tied to the Hodgkin-Huxley results on the behavior of gK and gNa. Since neuronal impulse encoding does not in general proceed in a space clamped region, the values of membrane potential and “leak” entering the differential equations are those at the trigger zone. The effects of electrotonic spread are then incorporated in the leak function. A particularly simple two parameter form of the model is explicitly written. By changing the value of one of the parameters the properties of the model change from that of exhibiting all features of rapid adaptation to those of tonic repetitive firing. Predictions of the model are discussed, as is the relationship between properties of the model and features of voltage clamp data.

Hotspots, mantle convection and plate tectonics: A synthetic calculation

A model is developed that unifies vigorous hotspots with global-scale mantle convection and plate tectonics. The convection dynamics are assumed to generate flow patterns that emerge as closely packed polygonal cells in approaching the asthenosphere, and whose geometry is completely determined by a defining set of vigorous hotspots. Overlying viscously coupled rigid plates are driven with unique velocities (Euler vectors) at which the area integral of the shear forces is zero; these velocities are dynamically stable. The computed plate velocities, resulting from convection based on 15 hotspots, are compared with the velocities of plate motion models AM1-2 (Minster andJordan, 1978) and HS-NUVEL1 (Gripp andGordon, 1990), which combine transform fault geometries, magnetic anomalies and seismic data. The comparison shows a striking agreement for a majority of the plates. Geophysical implications of this numerical exercise are discussed.

Electrical processes involved in the encoding of nerve impulses.

A mechanism for impulse encoding is advanced for those neurones whose impulse trigger zone membrane is more excitable than the general axonal membrane. Electrical communication between an electrotonically small patch of highly excitable membrane and neighboring membrane places the control of membrane potential — in varying degree — to the larger membrane area throughout the interspike intervals. That control is relinquished to the trigger membrane near the time of action potential initiation in a natural fashion. Model calculations demonstrate that this mechanism can lead to a dramatic lowering of the minimum stable firing frequency of tonic neurons, and, additionally influence the shape of the stimulus —versus — impulse frequency curve. The results are compared with the behavior of the slowly adapting stretch receptor neuron of the crayfish.

Voltage gating by molecular subunits of Na+ and K+ ion channels: higher-dimensional cubic kinetics, rate constants, and temperature

The structural similarity between the primary molecules of voltage-gated Na and K channels (alpha subunits) and activation gating in the Hodgkin-Huxley model is brought into full agreement by increasing the models sodium kinetics to fourth order (m(3) → m(4)). Both structures then virtually imply activation gating by four independent subprocesses acting in parallel. The kinetics coalesce in four-dimensional (4D) cubic diagrams (16 states, 32 reversible transitions) that show the structure to be highly failure resistant against significant partial loss of gating function. Rate constants, as fitted in phase plot data of retinal ganglion cell excitation, reflect the molecular nature of the gating transitions. Additional dimensions (6D cubic diagrams) accommodate kinetically coupled sodium inactivation and gating processes associated with beta subunits. The gating transitions of coupled sodium inactivation appear to be thermodynamically irreversible; response to dielectric surface charges (capacitive displacement) provides a potential energy source for those transitions and yields highly energy-efficient excitation. A comparison of temperature responses of the squid giant axon (apparently Arrhenius) and mammalian channel gating yields kinetic Q10 = 2.2 for alpha unit gating, whose transitions are rate-limiting at mammalian temperatures; beta unit kinetic Q10 = 14 reproduces the observed non-Arrhenius deviation of mammalian gating at low temperatures; the Q10 of sodium inactivation gating matches the rate-limiting component of activation gating at all temperatures. The model kinetics reproduce the physiologically large frequency range for repetitive firing in ganglion cells and the physiologically observed strong temperature dependence of recovery from inactivation.

Excitation parameters of the repetitive firing mechanism from a statistical evaluation of nerve impulse trains

Repetitive firing of single tonic neurones is modeled to include in detail both membrane excitation kinetics and electrotonic effects due to membrane non-uniformities in the impulse encoder region. The model is evaluated dynamically and compared with similar data obtained from the crayfish stretch receptor neuron. Two dynamic techniques utilizing small amplitude sinusoidal signals are employed. One technique is used to fix the values of two parameters which relate to the electrotonic control of membrane potential in the interspike interval and to the relaxation time of the K-conductance during repetitive firing. The other technique is employed as a consistency check. The dynamics are particularly sensitive to the K-channel relaxation time in the interspike interval.

Gating Kinetics of Stochastic Single K Channels

Recent papers by Conti and Neher (1980) and by Sigworth and Neher (1980) have shown that the conducting state of a single K+ or Na+ channel is “all-or-none”; that is, the channel is either open or closed. The actual occurrence of transitions between these two states appears to be a stochastic process. Nevertheless, the probability of finding a single channel in the open state is a function of voltage and time.

Adaptation and accommodation in the squid axon

Current clamp data of the squid axon indicate that there is a qualitative change in the adaptive response as the magnitude of the current step is increased. Large stimulus currents have a strong inhibitory effect on spike generation and on active responses in general. Such currents always lead to only one action-potential and to the elimination of post-spike subthreshold oscillation. In view of a direct connection between stimulus current and potassium current IK, the potassium channel of the Hodgkin-Huxley model is reinterpreted in a natural way such that the K+ conductance is directly dependent on IK in addition to a voltage dependence. The I-Kdependence seems to dominate whenever the stimulus current is greater than approximately 35 μA/cm2. For current ramps, and large current steps, such a current formulation leads to good agreement with the data.