Jürgen F. Fohlmeister, Principal Investigator

Computational Neuroscience

This research project involves the simultaneous solution of 9 to 14 non-linear differential equations for the purpose of determining the parameters of rate constants for the gating of 5 or 6 ionic channels. Furthermore, the channels were distributed non-uniformly throughout the complex geometry of the neuron membrane. The effects of this geometry on the encoding of nerve impulses were then evaluated.

This group also investigated the role of membrane capacitance in neural excitation. Although membrane capacitance (1 µF/cm2) was accurately measured about two decades prior to the first delineation of the ionic current responsible for the nerve impulse (the Hodgkin-Huxley model), capacitance of neural membrane has generally been regarded as an unavoidable and unimportant linear current path in parallel with the critical sodium, calcium, and potassium currents. More recently, however, it has become important to model impulse encoding in neurons of the central nervous system with non-uniform channel density distributions, for which the membrane capacitance can introduce unexpected excitation phenomena. This research group has shown that these phenomena are of two kinds, which are related to the charge-storage function of capacitance. In the space-clamped mode, capacitance has the important function of determining the scale for the magnitude of the ionic current necessary for impulse generation. In the non-space-clamped neuron, specifically in neurons with non-uniform channel density distributions, the capacitance can act as a battery in that stimulus current can charge the membrane capacitance, which is capable of holding its voltage under certain circumstances. This occurs when regions of low channel density are present in the neural morphology; the locally low charge leakage can control the dynamic range of impulse frequency generation. This is found specifically in retinal ganglion cells (as well as hippocampal neurons), where the distribution and density of channels on the dendrites becomes an important factor in impulse encoding, even when the dendrites themselves are incapable of supporting impulses on their membrane. Other specialized neural regions, specifically the impulse “trigger zone” which operates with the highest electrically gated channel densities, also interact with their neighboring membranes to determine rate of impulse firing.

This group also studied the Hodgkin-Huxley model equations. Although the Hodgkin- Huxley model for the squid axon membrane is incapable of generating nerve impulses at mammalian temperatures, the mathematical structure of the Hodgkin-Huxley (1952) equations have nevertheless been the basis for their simulation throughout the last half century at all physiological temperatures; this has been achieved by ad hoc adjustments to the equations. The researchers have discovered that the primary reason for the mammalian temperature failure of the Hodgkin-Huxley model lies in the magnitude of that model’s gating kinetic rate constants in the presence of membrane capacitance; in the absence of the capacitance the model will generate impulses at all temperatures. Applying the Q10 factor of 3 to the temperature dependence of all rate constants narrows the impulse from 2.5 ms (6.3 ˚C) to 0.6 ms (38 ˚C) in the absence of capacitative current. On the other hand, the impulse collapses for the same (and lesser) temperature shifts in the presence of capacitance. Phase space analysis shows that the repolarizing K-current overtakes the normally faster regenerative Na-current, because the rapid rate-of-rise (large dV/dt) of the Na-current is compromised by the necessarily accompanying large capacitative current, which is proportional to dV/dt; the slower recovering K-current is associated with a smaller time-rate of voltage change (dV/dt) and is therefore less affected by capacitative current. It is shown that phase space analysis offers the necessary adjustments to voltageclamp data, which are typically contaminated by artifacts due to the difficulty of achieving the necessary space-clamp, effects that are most pronounced in simulations involving large time-rates of change.

A final area of research involved creating a model, called the “sticky model,” which simulates the primordial accreting process of continents and their associated tectonic plates. Buoyant “flakes” of hard material are assumed to be persistently and randomly generated at isolated convective up-welling centers of the rocky Earth (mantle) and are driven horizontally and radially away from the up-welling centers (“hotspots”) on the upper surface of convection cells. The cells (Rayleigh-Benard convection) are defined by Voronoi polygons based on the distributed hotspots. The Voronoi edges represent convective down-welling regions, among which certain Voronoi triple junctions act as nucleation points for the accumulation of the buoyant material (flakes), which does not descend with the bulk mantle material. The accumulation represents the growth of the primordial continents, and associated tectonic plates. While flakes are moving, the nascent plates independently undergo rigid body motion imposed by the viscous shear forces exerted by the underlying convecting mantle. When a flake touches a plate it sticks to that plate and becomes a part of that plate, thus increasing the plate area. Tiny, aboriginal plates can be of arbitrary shape; their areas however grow much more rapidly along the Voronoi edges that elsewhere by the present model accretion process, resulting in starshaped proto-continents. When the plate area has grown to encroach upon, and cover the hotspots, the plate rotates erratically and begins to move away from its nucleation point, to wander over the global surface driven by the mantle convection cells. Plate-to-plate collisions occur, resulting in supercontinents, and the classical Wilson cycle of the accretion of supercontinents and their breakup will have begun.